Contributions to the Taxonomy of Rings

A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University

In partial fulfillment of the requirements for the degree Doctor of Philosophy

Henry Chimal-Dzul April 2021

© 2021 Henry Chimal-Dzul. All Rights Reserved. 2

This dissertation titled Contributions to the Taxonomy of Rings

by

HENRY CHIMAL-DZUL

has been approved for the Department of Mathematics and the College of Arts and Sciences by

Sergio R. L´opez-Permouth Professor of Mathematics

Florenz Plassmann Dean, College of Arts and Sciences 3 Abstract

CHIMAL-DZUL, HENRY, Ph.D., April 2021, Mathematics Contributions to the Taxonomy of Rings (139 pp.) Director of Dissertation: Sergio R. L´opez-Permouth The present work aims to contribute to the taxonomy of various classes of non-commutative rings. These include the classes of reversible, reflexive, semicommutative, 2-primal, NI, abelian, Dedekind finite, Armendariz, and McCoy rings. In addition, two new families of rings are introduced, namely right real McCoy and polynomial semicommutative. The first contribution concerns the hierarchy and interconnections of reflexive, abelian, and semicommutative rings in the setting of finite rings. It is shown that a minimal reflexive abelian non-semicommutative ring has order 256 and that the group algebra F2D8 is an example of such a ring. This answers an open question in a paper on a taxonomy of finite 2-primal rings. The second set of contributions includes characterizations of reflexive, 2-primal, weakly 2-primal and NI rings in the setting of Morita context rings. The results on Morita context rings which are reflexive are shown to generalize known characterizations of prime and semiprime Morita context rings. Similarly, the results on 2-primal, weakly 2-primal and NI rings Morita context rings presented here generalize various known results for upper triangular matrix rings. Specially, a characterization of NI Morita context rings is shown to be equivalent to the (most) famous conjecture in Ring Theory: K¨othe’s conjecture. The third collection of contributions addresses a study of right real McCoy rings and polynomial semicommutative rings; both classes of rings being introduced in this dissertation. The correlations between these types of rings is described, and a series of examples of finite polynomial semicommutative rings is given. Also, 4 equivalent conditions to the McCoy condition and some of its variations are also addressed. Finally, the last contribution establishes that the rings of column finite matrices, and row and column finite matrices over a reflexive ring are reflexive non-Dedekind finite. This result, along with some examples of finite non-reflexive Dedekind-finite rings that appear early in this dissertation, distinguishes the classes of reflexive and Dedekind-finite rings. 5 Dedication

To My Parents who worked tirelessly to foster every aspect of my education in my youth 6 Acknowledgments

I express my immense gratitude to my wife for constantly pushing me to challenge myself. Specially, I would like to thank her for her patience and support during this five years, and because she left a lot behind to undertake this journey with me in an unknown country. I would not be where I am now without her support. I express my heartfelt gratitude to my advisors, Professors Sergio L´opez-Permouth and Steve Szabo, for accepting me as his student and their constant support during these past five years. Because of their great sense of humor and insightful discussions in Mathematics, Sergio and Steve are persons who you instantly love and never forget once you meet them. I thank Professor Sergio for his exceptional Algebra courses full of “algebraic possums”, for his amazing Coding Theory courses that made me love that field, for his pictorial way to teach me Quasi-Frobenius rings, for his multiple “colochito” and “kishpinudo” examples, and for his translations from “Yucateco” to “Chap´ın”.More importantly, for introducing me to Professor Steve through the Center of Ring Theory and its Applications. I thank Professor Steve for introducing me to some central topics in this dissertation, for always being available to chat in Skype hundreds of times a week, for asking more questions than I could answer, and for his great and unique vision decomposing rings using idempotents. Finally, in Steve and Sergio, I have found not only two great advisors, but two best friends on which I can deeply trust and always consider as members of my family. I am very grateful for all what I have learned from you. I thank my great friends and parents in USA, Dr. Erik Boczko and Nickie Bailes or as I call them with great affection “The Malitos”. I thank Dr. Boczko for welcoming as his son in his house in January 2017 even I was a stranger to him. I 7 also thank him for his great discussions in Mathematics; for all his advise to succeed in my personal and professional life, and for sharing with me and my wife many adventures and UFC fights. I thank Nickie for always being there when I need help academically and personally. Their support made my life easier at Ohio University. To my Professors at Ohio University I owe my thanks for their wonderful lectures that strengthened my knowledge in Mathematics. Specially, to Professor Wei Lin for a great course in Statistics; Professor Archil Gullisashvili for his amazing lectures that made me to appreciate Real Analysis; Professor Vladimir Uspenskiy whose topolgy classes I was honored to attend; Professor Winfred Just for his outstanding lectures on Dynamical Systems and his interesting mathematical discussions inside and outside the classroom. Last but not least, to Ashley Pallone, Isaac Owusu-Mensah, Prabha Sherstha Shrestha, K. C. Rabi, and Rebi Muhammad (the “Big Big Big Boss”), for the privilege of sharing the 423 office at Morton Hall and all those big smiles that we had together. Also, I also must mention Aldo Garc´ıa-Guinto who became my little brother since the first day I met him at Ohio University. I will always remember each one of with great affection. 8 Table of Contents

Page

Abstract...... 3

Dedication...... 5

Acknowledgments...... 6

Introduction...... 10

1 On Rings and Zero-divisors...... 16 1.1 Various Classes of Rings...... 16 1.1.1 Symmetric Rings...... 19 1.1.2 Reversible Rings...... 20 1.1.3 Semicommutative Rings...... 22 1.1.4 Reflexive Rings...... 24 1.1.5 Duo Rings...... 26 1.2 Radicals and More Classes of Rings...... 28 1.2.1 PS -I and 2-primal Rings...... 32 1.2.2 Weakly 2-primal, NI and NR Rings...... 34 1.3 Diagrams of Implications...... 36

2 Minimal Reflexive non-Semicommutative Rings...... 40 2.1 Introduction...... 40 2.2 Finite Local, Reflexive and Semicommutative Rings...... 41 2.3 Small Abelian Reflexive Rings...... 45 2.4 Examples of Abelian Reflexive non-Semicommutative Rings..... 51

3 Morita Context Rings...... 61 3.1 Background on Morita Context Rings and Goals...... 61 3.2 Reflexive Morita Context Rings...... 63 3.2.1 Reflexive Morita Context Rings...... 64 3.2.2 Semiprime Morita Context Rings...... 70 3.2.3 Prime Morita Context Rings...... 73 3.3 NI Morita Context Rings...... 76 3.3.1 NI Morita Context Rings...... 77 3.3.2 Weakly 2-primal and 2-primal Cases...... 81 9

4 McCoy Rings and Polynomial Semicommutative Rings...... 84 4.1 An Overview on McCoy Rings...... 84 4.2 Right Real McCoy Rings...... 87 4.3 Abelian McCoy Rings...... 92 4.4 Polynomial Semicommutative Rings...... 95 4.5 McCoy Conditions and Polynomial Rings...... 102

5 Reflexive non-Dedekind-finite Rings...... 108 5.1 Row and Column Finite Matrices...... 109 5.2 The Universal Property of Monoid Rings...... 111 5.3 A Monoid Ring Embedded in RCFM(R)...... 113 5.4 Reflexive Rings and Column Finite Matrix Rings...... 124

References...... 129

Alphabetical Index...... 136 10 Introduction

The concepts of “zero-divisor” and “nilpotent element” play an important role in Ring Theory. Various types of rings have been defined to distinguish between rings satisfying certain properties related to zero divisors and nilpotent elements, which in some senses are generalizations of properties satisfied by commutative rings. These type of rings include reduced, symmetric, reversible, reflexive, semicommutative, 2-primal, weakly 2-primal, NI, Armendariz, and McCoy rings. In addition to these, we also have prime, semiprime rings (which are examples of reflexive rings), and right duo rings (which are instances of semicommutative rings). The hierarchy and interconnections between these various types of rings have been topics of intensive research (e.g. [1], [2], [3], [4], [5], [6]). On one hand, a goal in this dissertation is to contribute to the hierarchy and interconnections between the various types of rings mentioned above in the setting of finite rings. On the other hand, it aims to characterize various of the ring classes aforementioned in the setting of Morita context rings. Moreover, since various of the classes of interest here can be encompassed in two big categories, reflexive rings and Dedekind finite rings, examples of reflexive non-Dedekind-finite rings are constructed via row and column finite matrices. A brief review of the relevant work and motivations to this dissertation are next. The study of a taxonomy of 2-primal rings was initiated by Marks in [2], where 2-primal rings are analyzed in connection to reduced, symmetric, reversible, duo, semicommutative, and PS I rings. Marks produced various examples of rings to show that within the class of 2-primal rings, the other 6 ring properties are independent. All examples constructed in [2] are infinite rings (rings with an infinite number of elements), except one, which Marks created in response to a question 11 posed by Professor T. Y. Lam from University of California at Berkeley. Professor

Lam asked whether the group algebra F2Q8, where Q8 is the quaternion group, is a the smallest (with respect to order) finite reversible non-symmetric ring. Motivated by this question, Szabo [6] studied a taxonomy of 2-primal rings in the context of finite rings. In particular, Szabo constructed examples for the various possible combinations of ring classes related to finite 2-primal rings. Moreover, in [5] Szabo shows that the cardinality of a smallest reversible non-symmetric ring is 256, which gives a positive answer to Professor Lam’s question. In the papers [5] and [6], Szabo refers to finite rings with the smallest order satisfying certain ring properties as minimal rings. We follow this terminology in this document. An example of a minimal reflexive abelian non-semicommmutative ring proven to be more elusive in [5] and it was left for future consideration. Specifically, the question whether there is a finite local reflexive non-semicommutative ring is posed in [6]. It was observed in [6] that such a ring must have order at least 64 = 26. Thus this ring cannot be found by doing an exhaustive search among the current classification (up to isomorphism) of finite rings as it has been obtained only for finite rings of order pk where k ≤ 5 ( [7], [8]). The first contribution in this dissertation is to show that if p is a prime number and k < 8, then an abelian reflexive ring of order p8 is reversible, and so semicommutative. Moreover, it is proved that if D8 is the Dihedral group of order 8, then the group algebra F2D8 is a local reversible non-semicommutative ring.

Therefore, F2D8 is an example of a minimal reflexive non-semicommutative ring, which gives a solution to the question posed in [6]. Additional examples of finite local reversible non-semicommutative rings of minimal order are also provided in this dissertation, showing that F2D8 is not the only example of a such type of ring. 12

To construct examples of rings satisfying certain properties, it is natural to study the behaviour of classical ring constructions in relation to the ring properties of interest. Morita context rings have been used as a source to construct examples and counter-examples in Ring Theory. For instance, in [6] a minimal non-NI non-reflexive ring is proven to be isomorphic to the formal upper triangular matrix ring, which is a special case of a Morita ring:    F2 0 M ( )    2 F2    0  .  F2    0 F2

A minimal non-abelian NI reflexive finite ring has been shown in [6] to be the Morita context ring:   2 F2[x]/(x ) F2   .  2  F2 F2[x]/(x ) Similar constructions for infinite rings where used by Marks [2] in his study of a taxonomy of 2-primal rings. On the other hand, there are partial results concerning Morita context rings satisfying some of the ring properties studied here. For instance, necessary and sufficient conditions for an upper triangular matrix ring to be 2-primal have been investigated by Birkenmeier et al. [9]. Morita context rings which are prime or semiprime rings have also been characterized by Nicholson and Watter [10] and Hongan [11], respectively. Motivated by the examples of rings in [6] and [2], and the partial results on Morita context rings in [9], [10], and [11], characterization of reflexive, 2-primal, weakly 2-primal and NI Morita context rings are investigated in this dissertation. The characterization of reflexive Morita context rings generalizes the characterizations of prime and semiprime Morita context rings given by Nicholson 13

and Watter in [10], and Hongan in [11], respectively. Similarly, the characterizations of 2-primal, weakly 2-primal, and NI Morita context rings generalize the partial results given by Birkenmeier et. al in [9]. In addition to this, it is proven that a characterization of NI Morita context rings studied here is equivalent to the famous conjecture in ring theory known as K¨othe’sconjecture. McCoy rings have been recently introduced by Rege and Chhawchharia in [4], and by Nielsen in [3]. These rings are named in honor to N. H. McCoy who proved that for a commutative ring R, a zero divisor in R[x] is annihilated by a non-zero element in the ring R. McCoy rings has been investigated in relation to Armendariz, reduced, symmetric, duo, and semicommutative rings (e.g. [12], [1], [13], [14], [15], [16], [3], [4]). Furthermore, some variations of the definition of a McCoy ring has also been subject of research in connection with the five classes of rings aforementioned ( e.g. [1], [12], [15], [17]). In particular, the notions of Armendarizlike rings and strongly right McCoy rings have been introduced in [15] and [16], respectively, as sub-families of McCoy rings that encompass all reduced, right duo and Armendariz rings. Remarkably, not all semicommutative rings are McCoy as it was shown by Nielsen [3]. However, Hirano [14, Theorem 2.2] proved that if R[x] is semicommutative then R is a McCoy ring. It was observed by Huh et al. [18], that there is a semicommutative ring R such that R[x] is not semicommutative. Thus the class of rings which have a semicommutative polynomial ring deserves special attention. Such rings are called polynomial semicommutative in this dissertation. Also, a special case in the definition of a strongly right McCoy ring is also studied and called right real McCoy. It is shown that this notion offers a simpler view of Armendarizlike rings. As right real McCoy rings are strongly right McCoy, a natural question is whether this two classes of rings are different. It is shown here that 14

Example 1.9 in [17], which has been claimed to be an Armendarizlike non-strongly right McCoy ring, is erroneous. Unfortunately, we have not been able to provide an example of such ring, and we leave the question open. On the other hand, it is shown in this work that the class of abelian right McCoy rings properly contains the classes of reversible, right duo, Armendariz, and polynomial semicommutative rings. Examples of finite local polynomial semicommutative rings are presented, which show that a polynomial commutative ring may be or not at the same time reversible, duo, and Armendariz, or any combination of these three types of rings. Lastly, equivalent conditions to the ring properties of McCoy rings and its variations are contributed. As it was mentioned before, various of the classes of interest here can be encompassed in two big categories: reflexive rings and Dedekind finite rings. Finite rings are Dedekind-finite. Thus many of the examples of rings presented in this dissertation are Dedekind-finite. So, any finite non-reflexive ring serves as an example of a Dedekind-finite non-reflexive ring. However, as Noetherian rings are Dedekind-finite, no finite ring serves as an example of a non-Dedekind-finite ring. Non-Dedekind-finite rings are necessarily non-Noetherian. Many examples of rings are known to be Noetherian, but there are also plenty non-Noetherian rings. As a last set of contributions, in this dissertation it is shown that the ring CFM(R) of all column finite matrices, and the ring RCFM(R) of all row and column finite matrices, are reflexive if and only if the ring R is reflexive. Since RCFM(R) and CFM(R) are non-Dedekind-finite rings, this result lands a hand-full of examples of non-Dedekind-finite rings. The organization of this dissertation is as follows. In Chapter1, the definitions of the various classes of rings along with some preliminary results are given. Chapter2 focuses on showing that a minimal reflexive 15 non-semicommutative ring has order 256. Chapter3 presents the characterizations of Morita context rings which are reflexive, 2-primal, weakly 2-primal and NI. Chapter4 examines McCoy rings and the two new classes of rings introduced in this dissertation: right real McCoy rings and polynomial semicommutative rings. Finally, Chapter5 centers on showing that CFM(R) and RCFM(R) are examples of reflexive non-Dedekind finite rings provided that R is reflexive. It is worth to mention that the results in Chapters 2, 3, and 4 of this dissertation has been submitted for publication in the papers [19], [20], and [21], respectively. In particular, [19] has been recently published, while [20] is in the process of being published. 16 1 On Rings and Zero-divisors

The aim of this chapter is to introduce the definitions of the various classes of rings that form the core of this dissertation. Unless otherwise stated, throughout this work the word “ring” means an associative ring with identity 1 6= 0. Rings are not necessarily commutative. In fact, most of the ring properties that will be presented here are satisfied trivially by commutative rings. Hence, the focus of this dissertation is in non-commutative rings.

1.1 Various Classes of Rings

Let R be a ring. An element a ∈ R is said to be nilpotent if an = 0 for some integer n ≥ 1. A nonzero element a ∈ R is called a left zero-divisor if ab = 0 for some nonzero b ∈ R. The element b ∈ R is called a right zero-divisor. An element e ∈ R is called idempotent if e2 = e. An element x ∈ R is said to be central if rx = xr for all r ∈ R. Of course, for a commutative ring R, the concepts of left zero-divisor and right zero-divisor coincide, so we just speak of zero-divisors. Also, observe that the following ring properties trivially hold in the commutative setting: for all a, b, c ∈ R, 1. abc = 0 =⇒ bac = 0;

2. ab = 0 =⇒ ba = 0;

3. ab = 0 =⇒ aRb = 0;

4. aRb = 0 =⇒ bRa = 0; and

5. every left ideal is a (two-sided) ideal.

Remark 1. The expression aRb = 0 in (3) must be written as aRb = {0} because aRb = {arb : r ∈ R}. Similarly, bRa = 0 must be written as bRa = {0}. However, in 17

Algebra it is often to write 0 instead of {0}. We follow this abuse of notation in this document.

As one suspects, these five ring properties may be false for non-commutative rings. Some examples to illustrate these facts are given next.

Example 1.1. Let F be field and R = M3(F ) the ring of 3 × 3 matrices over F .

Let Eij be the matrix whose entries are all zero except the (i, j)-entry, which is 1 ∈ F . Then R does not satisfy the property “abc = 0 =⇒ bac = 0” because

E12E31E22 = 0 but E31E12E22 = E32 6= 0.

The ring R also does not satisfy the property “ab = 0 =⇒ ba = 0” because

E12E31 = 0 but E31E12 = E32 6= 0.

Similarly, since E12E31 = 0 but E12E23E31 = E11 6= 0, then R does not satisfy the property “ab = 0 =⇒ aRb = 0”.   FF   Example 1.2. Let F be a field and R =  , the ring of 2 × 2 upper 0 F     0 0 1 0     triangular matrices over F . Let a =   and b =  . Then for any 0 1 0 0   x y   r =   ∈ R, we have 0 z         0 0 x y 1 0 0 0         arb =       =   . 0 1 0 z 0 0 0 0

Hence aRb = 0. However, bRa 6= 0 because             1 0 x y 0 0 x y 0 0 0 y             bra =       =     =   . 0 0 0 z 0 1 0 0 0 1 0 0

Therefore R does not satisfy the property “aRb = 0 =⇒ bRa = 0”. 18

To illustrate that there are non-commutative rings that do not satisfy property (5) above, we consider again the ring in Example 1.2.   FF   Example 1.3. Let F be a field and R =  . It is easy to verify that 0 F   F 0   L =   is a left but not a right ideal of R. Thus L is not an ideal of R. 0 0

While there are various non-commutative rings that do not satisfy one or various of the ring properties (1)–(5), there are lots of non-commutative rings for which those implications are true. To present some examples, we recall the definitions of two standard classes of rings.

Definition 1.4. A domain is a ring without left and right zero-divisors. A ring is called reduced if it has no nonzero nilpotent elements.

A domain obviously satisfies the properties (1)–(4). Moreover, a domain is a reduced ring, but the converse implication is not true in general. For example, the ring R = Z × Z is a reduced ring which is not a domain ((1, 0) and (0, 1) are zero divisors). It is straightforward to verify that a ring is reduced if and only if, for all a ∈ R, a2 = 0 implies a = 0. Using this characterization, it can be shown that every reduced ring satisfies (1)–(4). For instance, if R is reduced and ab = 0 then (ba)2 = baba = 0, whence ba = 0, so R satisfies the property (2) above. The previous remarks motivate the definition of the following classes of rings.

Definition 1.5. A ring R is called: 1. symmetric if for all a, b, c ∈ R, abc = 0 =⇒ bac = 0.

2. reversible if for all a, b ∈ R, ab = 0 =⇒ ba = 0. 19

3. semicommutative if for all a, b ∈ R, ab = 0 =⇒ aRb = 0.

4. reflexive if for all a, b ∈ R, aRb = 0 =⇒ bRa = 0.

5. right duo (resp., left duo) if every right ideal (resp., left ideal) is an ideal.

In addition to the previous remarks, each one of the classes of rings has been related to interesting results in ring theory. A brief historical review and relevant work related to these classes of rings follows.

1.1.1 Symmetric Rings

Symmetric rings were first considered by Lambek [22], who showed the importance of these rings in the theory of representation of modules [22, 23]. Anderson and Camilo [24] used the term ZC3 to refer to a symmetric ring. Recall that a ring is symmetric if for a, b, c ∈ R, abc = 0 implies bac = 0. As we noted above, every reduced ring satisfies the properties (1)–(4). Hence every reduced ring is symmetric. This has been also noted by Anderson and Camilo in [24, Theorem I.3]. We record this remark in the form of a lemma.

Lemma 1.6 ([24]). A reduced ring is symmetric.

In the next example we provide a symmetric ring which is not reduced, showing the implication “Reduced =⇒ Symmetric” is irreversible.

Example 1.7. Let n ≥ 2 be an integer. The ring Zn of integers modulo n is a reduced ring if and only if n is square-free. To prove this, assume that the

α1 αk factorization of n as product of primes is n = p1 ··· pk . By the Chinese Remainder ∼ Theorem, n = α1 × · · · × α1 . Since αi has nilpotent elements if and only if Z Zp1 Zp1 Zpi

α1 ≥ 2, it follows that Zn is reduced if and only if αi = 1 for all i. Consequently, if n

is not square-free, then Zn is a symmetric non-reduced ring. 20

The following result on symmetric rings is a restatement of [24, Theorem I.1] and [22, Proposition 1 and Corollary 1].

Theorem 1.8 ([22, 24]). The following conditions are equivalent for a ring R. 1. R is symmetric.

2. for all n ≥ 3 and a1, . . . , an ∈ R, a1 ··· an = 0 implies aσ(1) ··· aσ(n) = 0 for

any permutation σ ∈ Sn.

3. ABC = {0} implies BAC = {0} for any nonempty subsets A, B, C of R, where XYZ = {xyz : x ∈ X, y ∈ Y, z ∈ Z} for any X,Y,Z ⊆ R.

4. A1A2 ··· An = {0} implies Aσ(1) ··· Aσ(n) = {0} for any nonempty subsets

A1,...,An of R and any permutation σ ∈ Sn.

Because the definition of symmetric rings is element-wise, the following is easy to verify.

Proposition 1.9. The class of symmetric rings is closed under subrings. Q Furthermore, for a family {Ri : i ∈ I} of rings, the direct product i∈I Ri is

symmetric if and only if Ri is symmetric for all i ∈ I.

Generalizations of the notion of symmetric rings and their relationship to other ring classes have been studied, for instance, in [25–31].

1.1.2 Reversible Rings

Reversible rings were introduced by P. M. Cohn in [32]. The purpose of Cohn in [32] was to study a class of rings which includes all commutative rings and domains. Using reversible rings, Cohn generalizes in [32] the classical result in commutative algebra, which states that an integral domain embeds in its quotient field. Anderson and Camilo [24] used the term CZ2 to refer to reversible rings. Recall that a ring R is reversible if for all a, b ∈ R, ab = 0 implies ba = 0. 21

Huh et al. [27] studied several properties of symmetric and reversible rings. Some properties of reversible rings are given next (see [33, Lemma 1]).

Proposition 1.10 ([33]). The class of reversible rings is closed under subrings. Q Furthermore, for a family {Ri : i ∈ I} of rings, i∈I Ri is reversible if and only if every Ri is reversible for all i ∈ I.

Marks [30] also studied the relation between reversible and symmetric rings with and without identity. He showed that for rings without identity, reversible and symmetric rings are independent classes of rings. However, for rings with identity as it is assumed in this dissertation, it is easy to see that a symmetric ring is reversible.

Lemma 1.11 ([30]). A symmetric ring is reversible.

An example of a reversible non-symmetric ring is given in [24, Example I.5]. Hence the implication “Symmetric =⇒ Reversible” is not reversible. Marks also provided in [30, Example 7] a finite ring which is reversible but not symmetric, which is given next.

Example 1.12 ([30]). The group algebra F2Q8, where Q8 = {±1, ±i, ±j, ±k} is the Quaternion group, is a reversible non-symmetric ring.

The previous example was constructed by Marks [30] in response to a question posed by Professor T. Y. Lam from University of California at Berkeley, who asked for a “small ring” (with respect to order) which is reversible non-symmetric.

However, at that moment, Marks did not know whether F2Q8 is a smallest ring ((with respect to order)) of such type. Motivated by this “funny little problem” (as Mark said), Szabo studied in [5] symmetric and reversible rings of the smallest order possible. Szabo refered to such rings as minimal rings. Of course, a search of minimal rings makes sense only in the context of finite rings. The main result of [5] 22 states that F2Q8 is a minimal reversible non-symmetric ring. Due to the importance of this result for the purposes of this work, we record this remark in the following:

Proposition 1.13 ([5]). The group algebra F2Q8 is a minimal reversible non-symmetric ring.

We now give an important property of finite reversible rings due to Szabo [5, Proposition 2.3]. Recall that a ring R is indecomposable if R cannot be written as ∼ R = R1 × R2 for non-zero rings R1 and R2. A ring R is said to be local if it has only one maximal ideal.

Proposition 1.14 ([5]). A finite reversible ring is local if and only if it is indecomposable.

Further results on reversible rings can be found, for instance, in [5,6, 17, 27, 30, 33–35].

1.1.3 Semicommutative Rings

The study of semicommutative rings was initiated by Bell in [36], where semicommutative rings are referred to as rings having the Insertion-of-Factors Property or IFP rings. Shin [37] used the name (S I) for semicommutative rings. The name of zero-insertive ring has been also used to refer to a semicommutative ring in [38]. The name “semicommutative ring” was coined by Motais de Narbonne in [39]. Recall that a ring R is semicommmutative if for all a, b ∈ R, ab = 0 implies aRb = 0. Recall also that for an element a ∈ R, the left annihilator of a is defined as `.Ann(a) = {r ∈ R : ra = 0}. Simmirally, the right annihilator of a is defined as r.Ann(a) = {r ∈ R : ar = 0}. It is easy to see from these definitions that the left annihilator is a left ideal of R, while the right annihilator is a right ideal of R. In 23 the following proposition we quote a result due to Shin (see [37, Lemma 1.2]), which can be proved directly from the definition of a semicommutative ring and connects the definition of a semicommmutative ring with the right and left annihilator of elements in the ring.

Proposition 1.15 ([37]). Let R be a ring. The following are equivalent: 1. R is semicommutative.

2. For each element a ∈ R, r.Ann(a) is an ideal of R.

3. For each element a ∈ R, `.Ann(a) is an ideal of R.

Shin also proved in [37, Proposition 1.4] that a symmetric ring is semicommutative. Since a symmetric ring is reversible by Lemma 1.11, it is natural to think that a reversible ring is also semicommutative. We answer this in the affirmative in the next lemma.

Lemma 1.16. A reversible ring is semicommutative.

Proof. Assume R is a reversible ring. Let a, b ∈ R and suppose that ab = 0. Then, for all r ∈ R, bar = 0, so arb = 0 by the reversibility of R. Therefore aRb = 0, i.e., R is semicommutative.

The structure of minimal non-commutative semicommutative rings has been obtained by Xu [38]. In [38, Theorem 8], it is shown that such a ring has order 16. One of the rings given by Xu in [38] is the following, which also shows that the class of semicommutative rings properly contains the class of reversible rings.

Example 1.17. Let F2hu, vi be the ring of polynomias in non-commuting

2 2 indeterminates u, v over the finite field F2. Let (v , vu, u − uv) be the ideal of

2 2 F2hu, vi generated by {v , vu, u − uv}. Consider the ring hu, vi R = F2 . (v2, vu, u2 − uv) 24

Then R is a semicommutative ring by [38, Example 7] (see also [6, Proposition 2.13]). Clearly, R is not reversible since vu = 0 but uv 6= 0.

The following properties of semicommutative rings are easy to verify (see also [37, Proposition 2.2 and Theorem 2.3]).

Proposition 1.18 ([37]). The class of semicommutative rings is closed under Q subrings. Furthermore, for a family {Ri : i ∈ I} of rings, the direct product i∈I Ri is semicommutative if and only if every Ri is semicommutative for all i ∈ I.

Additional properties of semicommutative rings have been investigated in [2,6, 37, 40]. Some generalizations of semicommutative rings can be found, for instance, in [41, 42].

1.1.4 Reflexive Rings

The concept of reflexive ring comes from the notion of reflexive ideal introduced by Mason [43]. According to Mason [43], an ideal I of a ring is called reflexive if, for all a, b ∈ R, aRb ⊆ I implies bRa ⊆ I. Mason [43], called a ring R reflexive if the zero ideal is reflexive, i.e., for all a, b ∈ R, aRb = 0 implies bRa = 0. Reflexive rings can be thought as a generalization of reversible and semiprime rings. Recall that a ring R is said to be semiprime if xRx = 0 implies x = 0.

Proposition 1.19. Reversible and semiprime rings are reflexive.

Proof. Assume R is reversible. Let a, b ∈ R such that aRb = 0 then ab = 0 and so (ra)b = 0 for all r ∈ R. Because R is reversible, bra = 0 showing bRa = 0. Thus R is reflexive. Assume now that R is a semiprime ring. Let a, b ∈ R such that aRb = 0. Then for every r ∈ R,(bra)R(bra) = br(aRb)ra = 0. Since R is semiprime, bra = 0 for all r ∈ R. This shows bRa = 0, so R is reflexive. 25

The previous proposition gives many examples of reflexive rings. One may suspect that semicommutative rings are also reflexive. However, the following example erases that posibility.

Example 1.20. Let S be a reduced ring. Consider the ring     a b c        R = 0 a d : a, b, c, d ∈ S .        0 0 a 

By [17, Proposition 1.2] R is semicommutative. However, by [44, Example 2.1], the ring R is not reflexive.

We also can show that reflexive rings are not necessarily semicommutative, where we conclude that the classes of reflexive and semicommutative rings are independent.

Example 1.21. Let F be any field. Since the ring Mn(F ) of n × n matrices over F is a semiprime ring, Mn(F ) is a reflexive ring for all n ≥ 2. However, for n ≥ 3, it can be shown as in Example 1.1 that Mn(F ) is not semicommutative.

Despite the independence between the classes of reflexive and semicommutative rings, we have the next result, which is [45, Proposition 2.2].

Proposition 1.22 ([45]). A ring R is reversible if and only if it is semicommutative and reflexive.

The following properties of reflexive rings are [33, Lemma 1 and Lemma 4].

Proposition 1.23 ([33]). Let {Ri : i ∈ I} be a family of rings. 1. The class of reflexive rings is not closed under subrings.

2. The following are equivalent: 26

(a) Ri is reflexive for all i ∈ I.

(b) The direct product of Ri is reflexive.

(c) The direct sum (possible without identity) of Ri is reflexive.

The next proposition, due to Kwak and Lee [45], implies that the reflexive property is Morita invariant (see [46] for more details). Recall that an element e ∈ R, where R is a ring, is called idempotent if e2 = e.

Proposition 1.24 ([45]). Let R be a ring and e ∈ R be an idempotent.

1. If R is reflexive then eRe is also reflexive.

2. R is reflexive if and only if Mn(R) is reflexive for all n ≥ 1.

Various additional characterizations and properties of reflexive rings have been investigated by Kwak and Lee [45]. There, connections to semiprime rings, quasi-Baer rings and what they call idempotent reflexive rings were established. Kim and Lee [33] also studied some properties of reflexive rings.

1.1.5 Duo Rings

Recall that a ring R is called right duo if every right ideal is an ideal. A ring R is said to be left duo if every left ideal is an ideal. A ring that is both left and right duo is a duo ring. Duo rings were introduced by E. H. Feller [47] and represent a natural generalization of commutative rings. An equivalent formulation for a ring to be right or left duo is given next.

Proposition 1.25. For any ring R, the following statements are equivalent:

1. R is right duo (resp. left duo). 27

2. For all a, b ∈ R, ba ∈ aR (resp., ba ∈ Rb).

One-sided duo rings are examples of semicommutative rings.

Proposition 1.26. A right or left duo ring is semicommutative.

Proof. Assume R is right duo. For every a ∈ R, the right annihilator r.AnnR(a) is a right ideal of R. Since R is right duo, then r.Ann(a) is an ideal of R, whence R is semicommutative by Proposition 1.15.

The right-left distinction in the definition of duo rings is necessary as there are right duo but not left duo rings, and the other way around. The following example, which is [48, Example 3], illustrates this fact.

Example 1.27. Let F (x) be the rational function field over a field F . The field F (x) is isomorphic to F (x2) via the ring isomorphism σ : F (x) → F (x2) defined as

2 σ(f(x)) = f(x ). Consider the subring of M2(F (x)):     a 0    R =   : a, b ∈ F (x) .  b σ(a)    0 0   The ring R has only two right ideals: R and  . Clearly, R is a two-sided F (x) 0   0 0   ideal. Moreover, it is easy to check that   is in fact an ideal. However, F (x) 0   0 0   the left ideal   contains no nonzero ideal. Thus R is not left duo. F (x2) 0

In light of [49, Proposition 3], finite right duo rings and finite left duo rings coincide.

Proposition 1.28 ([49]). A finite right or left duo ring is a duo ring. 28

For the class of finite rings, it is natural to ask for minimal examples of non-commutative duo rings. Xue [50] determined the structure of such rings and showed that a minimal non-commutative duo ring is local and of order 16. An example of such ring is the finite ring (c.f. [50, Example 2]):

hu, vi F2 . (u2 − uv, v2 − uv, vu)

In [6], Szabo also studied minimal examples of duo rings which satisfy some of the additional ring properties being studied here. Further works concerning duo rings and some generalizations of their generalizations include [31, 47–54].

1.2 Radicals and More Classes of Rings

Let R be a ring and N(R) the set of all nilpotent elements of R, that is,

n N(R) = {a ∈ R : a = 0 for some n ∈ N}.

In commutative algebra, it is known that N(R) is an ideal of R (c.f. [55, √ Proposition 1.7]). This ideal is called the nilradical, and it is often denoted by 0 or Nil(R). The nilradical of a commutative ring can be characterized as the intersection of all prime ideals of R (see [55, Proposition 1.8]). Thus, in addition to the properties (1)–(5) above, the following is another important property satisfied by a commutative ring R:

6. N(R) = {a ∈ R : a is nilpotent} is an ideal of R.

However, as we mentioned before, the class of rings is much larger and one expects this property to fail for non-commutative rings.     0 1 0 0     Example 1.29. Let F be a field and R = M2(F ). Then   and   are 0 0 1 0 nilpotent elements, but their sum is a unit in R of order two. Hence N(R) is not an ideal of R. Moreover, notice that N(R) is not even a subgroup of (R, +). 29

Among other things, the pathology for non-commutative rings observed in the previous example have lead mathematicians to extend the theory of radicals for non-commutative rings. In what follows we briefly discuss the definitions and essential properties of four kinds of radicals that can be associated to a ring. We refer the reader to [56, §10 and §4] for more details concerning this material. Recall that a one-sided or two-sided ideal I of a ring R is said to be nilpotent if In = 0 for some integer n ≥ 1; while I is called nil if I ⊆ N(R). The ideal I is called

locally nilpotent if, for any finite subset S = {s1, s2, . . . , sn} of I, there exists an integer N = N(S, n) such that any product of N elements of S is zero. In other words, I is locally nilpotent if any subring (without identity) of I generated by a finite subset is a nilpotent ring without identity (A ring is called nilpotent if all its elements are nilpotent). The relation between these notions is as follows for a one-sided ideal I:

I is nilpotent I is locally nilpotent I is nil.

An ideal P of a ring R is said to be prime if P 6= R and, for ideals I,J of R,

IJ ⊆ P =⇒ I ⊆ P or J ⊆ P.

If the previous condition holds for all ideals I = J, then P is called a semiprime ideal. Clearly, a prime ideal is semiprime but the converse is not true in general. The sum of any family of one-sided locally nilpotent ideals is always a (two-sided) locally nilpotent. Also, the sum of any family of (two-sided) nil ideals is always nil. The intersection of any family of prime ideals is not always prime but it is semiprime.

Definition 1.30. Let R be a ring. The (Baer’s) lower nilradical or the

Baer-McCoy radical or the prime radical of R, denoted Nil∗(R), is defined as:

Nil∗(R) = the intersection of all prime ideals of R. 30

The Levitzki radical of R is defined as:

L(R) = the sum of all locally nilpotent ideals of R.

The upper nilradical of R is defined as

Nil∗(R) = the sum of all nil ideals of R.

The Jacobson radical of R is defined to be

J(R) = the intersection of all maximal left ideals of R.

Clearly, the prime nilradical is the equivalent version to the nilradical of a commutative ring. Although, its definition was simply stated in Definition 1.30, there are various concepts that needed to be developed before such statement can be made (see [56, §10]). The relationships between the radical in Definition 1.30 is as follows. It can be shown that Nil∗(R) is the smallest semiprime ideal of R, i.e., any semiprime ideal of

R contains Nil∗(R). The Levitzki radical is the largest locally nilpotent ideal of R. Also, observe that if I2 ⊆ L(R), then I2 is locally nilpotent, and so I is locally nilpotent. Consequently, L(R) is a semiprime ideal of R. Clearly, the upper nilradical is the largest nil ideal of R. Lastly, every left or right nil ideal is a subset of the Jacobson radical. Thus we have the following chain of inclusions:

∗ Nil∗(R) ⊆ L(R) ⊆ Nil (R) ⊆ J(R) ∩ N(R). (1.2.1)

If R is commutative then

∗ Nil∗(R) = L(R) = Nil (R) = N(R) ⊆ J(R).

If R is a left or right Artinian ring then

∗ Nil∗(R) = L(R) = Nil (R) = J(R) ⊆ N(R). 31

Moreover, the remarkable Levitzki’s Theorem (c.f. [56, Theorem 10.30]), asserts that for a Noetherian ring R:

∗ Nil∗(R) = L(R) = Nil (R) ⊆ J(R) ∩ N(R).

However, in general for any ring R, it is claimed in [56, pp. 177] that the inclusions in (1.2.1) are strict inclusions. Although, only two examples are provided in [56]

∗ ∗ showing that Nil (R) ( J(R) and Nil∗(R) ( Nil (R). One of the main contributions of [57] shows that the other inclusions are in fact strict. It is worth to note that the paper [57] is an independent work by Prof. Steve Szabo and myself that is not reported as part of this dissertation.

∗ The fact that in the commutative setting, Nil∗(R) = L(R) = Nil (R) = N(R), motivates the following definitions of classes of rings, to which we also add two other classes related to them.

Definition 1.31. A ring R is called:

7. 2-primal if Nil∗(R) = N(R).

8. weakly 2-primal if L(R) = N(R).

9. NI if Nil∗(R) = N(R).

10. NR if N(R) is an abelian group closed under multiplication.

11. PS I if for every a ∈ R, R/r.Ann(aR) is 2-primal.

It is clear from the previous discussions and the definitions that the following chain of implications hold for any ring:

PS I 2-primal weakly 2-primal NI NR

In the commutative setting all these classes coincide. In addition, in view of Levitzki’s Theorem, for Noetherian rings the following implications and equivalent 32

conditions hold for any ring:

PS I 2-primal weakly 2-primal NI NR

In particular, the previous implications hold for any finite ring. Moreover, for the class of finite rings, it has been shown in [6, Lemma 2.2] that the families of PS I rings and 2-primal coincide. Thus for the class of finite rings, the following is true.

PS I 2-primal weakly 2-primal NI NR

A brief review of the classes of PS I, 2-primal, weakly 2-primal, NI and NR rings is given in the next subsections.

1.2.1 PS -I and 2-primal Rings

The class of 2-primal rings was introduced by G. Shin [37], though the term “2-primal” was coined later by Birkenmeier, Heatherly and Lee in [58]. In [58, Section 2] a review of the historical origins of 2-primal rings is given. Influenced by Lambek’s investigation on symmetric rings [22], Shin’s main focus in [37] was on producing sheaf representations for a class of rings, which he named pseudo symmetric. According to Shin, a ring R is called pseudo symmetric if it satisfies two conditions, which he called PS I and PS II (see [37, pp.45]). The PS I ring property in Shin’s paper [37] is what have given place to PS I rings. In [37, Theorem 1.5], Shin proves that the set of nilpotent elements of a semicommutative ring coincides with its prime nilradical, i.e., a semicommutative ring is 2-primal. Using this fact, he showed in [37, Proposition 1.6(a)] the following, which connects the classes of rings in Section 1.1 with the classes of rings being studied in this section:

Lemma 1.32 ([37]). A semicommutative ring is PS I. 33

The converse of the implication “semicommutative =⇒ PS I” does not hold in general. An example of a PS I non-semicommutative ring is given in [2, Example 3.12]. It is worth to mention that [2, Example 3.12] is a ring with an infinite number of elements. A finite PS I non-semicommutative ring has been proved to be the

group ring F2D8 in [57, Example 3.1], where D8 is the Dihedral group of order 8. The following are equivalent formulations for a ring to be 2-primal. First, recall that an ideal I of a ring R is called completely prime if ab ∈ I implies a ∈ I or b ∈ I; and completely semiprime if a2 ∈ I implies that a ∈ I.

Proposition 1.33 ([37]). The following are equivalent for a ring R: 1. R is 2-primal.

2. Every minimal prime ideal is completely prime.

3. R/Nil∗(R) is reduced.

Among fundamental properties of 2-primal rings are the following.

Proposition 1.34 ([59]). Let {Ri : i ∈ I} be a family of rings.

1. Any subring (possibly without identity) of a 2-primal ring is 2-primal.

L 2. If Ri is 2-primal for all i ∈ I then the direct sum i∈I Ri is 2-primal.

3. If X is any set of commuting indeterminates, which commute with with every element of R, then R[X] is 2-primal.

It has been shown that the direct product of an arbitrary family of 2-primal rings is not necessarily 2-primal in [9, Example 1.6] or [60, Example 1]. Among various properties, Birkenmeier et al. [58] provided necessary and sufficient conditions for a formal upper triangular matrix ring to be 2-primal. 34

Proposition 1.35 ([58]). Let A and B be ring and M be an (A, B)-bimodule. Then   AM   the formal upper triangular matrix ring   is 2-primal if and only if A and 0 B B are 2-primal rings.

Further properties and results concerning 2-primal rings can be found, e.g., in [2,6,9, 57, 58, 60–63].

1.2.2 Weakly 2-primal, NI and NR Rings

The class of weakly 2-primal rings was introduced recently by Chen and Cui [64]. It is shown in [64] that weakly 2-primal rings are a family of rings for which some known results on semicommutative rings extend naturally. Specifically, we are refering to part (2) in the next proposition.

Proposition 1.36 ([64]). A subring of a weakly 2-primal ring is weakly 2-primal. Furthermore, 1. if R is weakly 2-primal, then the ring of n × n upper triangular matrices over R is weakly 2-primal;

2. if R is weakly 2-primal and α is an endomorphism of R satisfying the condition “ab = 0 if and only if aα(b) = 0 for all a, b ∈ R”, then the skew polynomial ring R[x; α] is weakly 2-primal.

It is also shown in [64, Example 3.1] that there is a weakly 2-primal ring that is not 2-primal. Hence the implication “ 2-primal =⇒ weakly 2-primal” is in general not reversible. In contrast to 2-primal rings, the class of weakly 2-primal rings is new and has not been widely explored. Hence, there are few works on weakly 2-primal rings. Some of these include [65–69]. In Chapter3 we contribute to the theory of weakly 2-primal rings in the setting of Morita context rings. 35

NI rings were introduced by Marks in [62]. Recall that a ring R is NI if and only if Nil∗(R) = N(R). The next result provides equivalent formulations for a ring R to be NI.

Proposition 1.37 ([59]). The following statements are equivalent. 1. R is NI.

2. N(R) is an ideal of R.

3. Nil∗(R) is a completely prime ideal.

4. R/Nil∗(R) is a reduced ring.

5. Every subring (possibly without identity) of R is NR

There are NI rings which are not 2-primal by [62, Example 2.2]. Hence the converse of the implication “2-primal =⇒ NI” does not hold in general. In [64], it is claimed that the implication “weakly 2-primal ⇒ NI” is also irreversible. However, none of the examples cited there show this fact. In [57, Example 3.12] an NI non-weakly 2-primal ring is constructed. This example is one of the main contributions in [57]. The literature concerning NI rings is vast. The paper [59] contains various results and constructions of NI rings. We also refer to the papers [70–75] for more information on NI rings . In Chapter3, we contribute to the theory of NI rings in the setting of Morita rings. The study of NR rings was inaugurated by Chun et al. in [70] as a generalization of Armendariz rings and NI rings (Armendariz rings will be defined later in Section 4). Recall that a ring R is NR if N(R) is a subring (without identity) of R. The following are fundamental properties of NR rings. 36

Proposition 1.38 ([70]). The class of NR rings is closed under subrings and direct sums. Moreover,

1. For any ring R, Mn(R) is not an NR ring.

2. If R/I is NR, where I is a nil ideal, then R is NR.

3. If R is NR and e ∈ R is an idempotent then er − re ∈ N(R) for all r ∈ R

Proposition 1.39 ([70]). Let A, B rings and M be an (A, B)-bimodule. Then the   AM   formal upper triangular matrix ring   is NR if and only if A and B are 0 B NR.

Although, NR rings are not studied in this dissertation, the purpose of mentioning their definition and fundamental properties here is to reference them in some concluding remarks in Chapter3. More information on NR rings can be found in [70, 76–78].

1.3 Diagrams of Implications

The purpose of this section is to summarize the relationships between the various ring classes that we have defined so far in the form of a diagram. To those classes of rings, we add the families of abelian and Dedekind-finite rings which we define below. A Dedekind-finite ring is also known as Directly-finite in the literature. Let R be a ring. An element x ∈ R is said to be central if xr = rx for all r ∈ R. That is, x ∈ R is central if it is in the center of the ring:

C(R) = {x ∈ R : xr = rx for all r ∈ R}.

Definition 1.40. A ring R is called: 1. abelian if every idempotent in R is central.

2. Dedekind-finite if, for all a, b ∈ R, ab = 1 implies ba = 1. 37

Abelian and Dedekind-finite rings are classes of rings with the longest history among the classes of rings studied here. Both types of rings are natural generalizations of commutative rings. Of course, not every ring is abelian or Dedekind-finite. We present some examples to justify this claim.   1 0   Example 1.41. Let F be a field and R = M2(F ). Then E11 =   is a 0 0 non-central idempotent. Hence R is non-abelian. Although, it is well-known from Linear Algebra, that R is Dedekind-finite.

Example 1.42. Let F be a field and V a vector space of countable infinite dimension over F . Let {ei : i ∈ N} be a basis of V . Let R = EndF (V ) be the ring of all vector space endomorphisms of V . Define a, b ∈ R as follows:

a(e1) = 0, a(ei) = ei−1 for all i ≥ 2, and

b(ei) = ei+1 for all i ≥ 1.

Then ab = 1 but ba 6= 1, so R is not Dedekind-finite.

The implications between the classes of semicommutative, abelian, NR and Dedekind-finite rings is given next.

Lemma 1.43. Semicommutative rings are abelian. Moreover, abelian and NR rings are Dedekind-finite.

Proof. Semicommutative rings are abelian by [36, Lemma 2(c)] or [37, Lemma 2.7]. Abelian rings are Dedekind-finite by [61, Proposition 2.7]. NR rings are Dedekind-finite by [70, Proposition 1.4].

In Figure 1.1, we summarize the implications between the various ring classes considered here. 38

Domain Reduced

Commutative Symmetric

Reversible Reflexive

Right duo Semicommutative

PS I

2-primal Abelian

Weakly 2-primal

NI NR Dedekind-finite Figure 1.1: Ring class implications

Recall that for a Noetherian ring, the classes of 2-primal, weakly 2-primal and NI coincide. Moreover, by [56, Exercise 1.12], a Noetherian ring is Dedekind-finite, so Figure 1.1 simplifies to the diagram in Figure 1.2. We can simplify the diagram in Figure 1.2 further for finite rings, by noticing the following result.

Lemma 1.44 ([6]). Finite reduced rings are commutative. Abelian rings are NI.

The series of examples presented so far indicates that no further implications hold for any of these diagrams, other than by transitivity and the one indicated in Proposition 1.22. 39

Domain Reduced

Commutative Symmetric

Reversible Reflexive

Rightduo Semicommutative Abelian

PSI

2 − primal NR Dedekind − finite Figure 1.2: Ring class implications for Noetherian rings

F ield

Reduced

Commutative Symmetric

Reversible Reflexive

Duo Semicommutative Abelian 2 − primal NR Figure 1.3: Ring class implications for finite rings 40 2 Minimal Reflexive non-Semicommutative Rings

2.1 Introduction

Minimal examples of various types of finite rings have become important in many fields of pure and applied mathematics. As an example, the ring U2(F2) is a ring of order 8 which is a minimal non-commutative ring. This ring is NI but notice that M2(F2) is not. This is actually a minimal non-NI ring (see [6]). Minimal examples of duo, reversible, semicommutative and Frobenius non-chain rings have been investigated by many authors (e.g. [33, 38, 79]). Minimal non-commutative reflexive rings has been studied by Kim and Lee [33]. Many of these minimal rings mentioned have order 16. In fact, it is shown in [33] that a minimal non-commutative reflexive ring has order 16 and that it is isomorphic to either M2(F2) or the ring     a b      : a, b ∈ F4 .  0 a2  In [80] it was shown that there are only 13 non-commutative rings of order 16. From their list and the fact that there is only one non-commutative ring of order 8, with some effort, the minimal rings mentioned so far could be found by simply sifting through this handful of rings. Minimal reflexive and non-reflexive finite rings, which satisfy additional ring properties like semicommutativity, have been studied by Szabo [6]. A minimal reflexive abelian non-semicommutative ring was the only type of ring that was left for future consideration in [6]. Although, it has been shown in [2, Lemma 5.3] that an infinite ring of such type does exist. From [6, Proposition 3.3], it is known that the order of such a ring is at least 64 = 26. Finding minimal rings is more difficult when rings of larger orders are involved since rings of order pn have only been fully classified for n ≤ 5 (see [7] and [8]). The 41

classification shows there are a few hundred rings of order p5, so it is clear that there is an abundance of rings of larger orders. So, without a classification of these higher orders it is cumbersome to find minimal rings. The purpose in this chapter is to show that a finite abelian reflexive ring of order pk for some prime p and k < 8 is reversible (Theorem 2.12). With this and the ring F2D8 in hand, which will be shown to be reflexive nonsemicommutative ring of order 256 (Example 2.13), it can be seen that a minimal abelian reflexive nonsemicommutative ring is of order 256 = 28 (Theorem 2.14). In Section 2.2 some basic results on finite rings and finite reflexive rings are given. Section 2.3 has the main result which shows that a finite abelian reflexive ring of order pk for some prime p and k < 8 is reversible. Section 2.4 shows that a minimal abelian reflexive nonsemicommutative ring has order 256. Examples of such minimal rings are also given in Section 2.4.

2.2 Finite Local, Reflexive and Semicommutative Rings

Recall that a finite ring is uniquely expressible as direct sum of rings of prime power order for distinct primes (see [81] for more information on finite rings). In light of Proposition 1.14, a reversible ring (and so reflexive) is local if and only if it is indecomposable. Moreover, since the direct sum of a collection of reflexive non-semicommutative rings is reflexive non-semicommutative, only finite local rings of order pn need to be considered. For basic facts about local rings, the reader may consult [56]. Here some important facts for the purposes of this dissertation are reviewed. When R is finite, J = J(R) is a nilpotent (two-sided) ideal, so there exists a least integer K such that J K+1 = 0 but J K 6= 0. The integer K + 1 is called the index of nilpotency of J. The following is a well-known result about finite local rings. 42

Lemma 2.1. Let R be a finite local ring, K + 1 the index of nilpotency of J = J(R) and F = R/J. Then F is a finite field and J i/J i+1 is a finite dimensional F -vector space for each 1 ≤ i ≤ K.

Using Lemma 2.1, it can be shown that in a local ring R with J = J(R), K + 1 the index of nilpotency of J and F = R/J, there exists a minimal set of generators

i i i+1 of J(R) , {ui1, . . . , uidi } ⊂ J \ J for each 1 ≤ i ≤ K, and that each element of R can be uniquely expressed in the form

K X α0 + αi1ui1 + ··· + αidi uidi , i=1

i i+1 where α0, αij ∈ F and di = dimF J /J . This fact will be used freely throughout the work. Recall that a ring R is called a left (resp. right) chain ring if its lattice of left (resp. right) ideals form a chain under inclusion. Finite chain rings have been characterized by E. W. Clark and D. A. Drake in [82, Lemma 1]. Pertinent facts on finite chain rings are collected in the following lemma.

Lemma 2.2. For a finite ring R with J = J(R), the following are equivalent: 1. R is a left chain ring.

2. R is a local ring and J is principal.

3. There exists u ∈ R such that every proper left or right ideal has the form

J i = huii for some i ∈ N.

4. R is a right chain ring.

Proposition 2.3. A finite chain ring is symmetric and duo.

Proof. Let R be a finite chain ring and J = J(R). By Lemma 2.2, there exists u ∈ J such that J = hui. For notational convenience set u0 = 1. Let x, y ∈ R and U(R) be 43

the group of units of R. Then there exist s1, s2, t1, t2 ∈ U(R) and α, β ≥ 0 such that

α β α α β β x = s1u , y = t1u , u t1 = t2u , and u s1 = s2u . So,

β α β+α −1 −1 β+α −1 −1 yx = (t1u )(s1u ) = t1s2u = t1s2t2 s1 s1t2u = t1s2t2 s1 xy.

This shows that yx is a left unit multiple of xy. We now prove that R is symmetric. Let a, b, c ∈ R such that abc = 0. By the arguments above, ba = ωab for ω ∈ U(R). Then bac = ωabc = 0. Therefore R is symmetric. Finally, R is duo by Lemma 2.2(3).

To conclude this section, three technical lemmas are given which are needed in the results of Section 2.3. Lemma 2.4 establishes that for a finite local ring R, if J(R)i is principal and J(R)i+1 6= 0 for some i, then J(R)j is principal for j ≥ i and that there is an element of J(R) \ J(R)2 that has the same index of nilpotency as J(R).

Lemma 2.4. Let R be a finite local ring, J = J(R) and F = R/J. Assume there is n ≥ 1 such that J n is principal and J n+1 6= 0. Then there is u ∈ J \ J 2 such that J l = huli for all l ≥ n.

2 n Proof. Let u1, . . . , us ∈ J \ J be such that J = hu1, . . . , usi. Since J is principal,

2 n there is x1, . . . , xn ∈ J \ J such that J = hx1 ··· xni. Hence

n+1 n+1 J = hu1x1 ··· xn, . . . , usx1 ··· xni. Also, by the fact that J 6= 0, we obtain

n+1 n+2 n+1 n+2 J 6= J , so that there is j such that ujx1 ··· xn ∈ J \ J . Let u = uj.

n n+1 Using that J /J is a 1-dimensional F -vector space, there is α1 ∈ F \{0} such that

n+1 n+1 n+1 ux1 ··· xn−1 + J = α1x1 ··· xn + J 6= J .

Then

2 n+2 n+2 n+2 u x1 ··· xn−1 + J = uα1x1 ··· xn + J 6= J . 44

n+1 n+2 Since α1 is a unit, uα1x1 ··· xn ∈ J \ J by the choice of u. Thus,

2 n+1 n+2 2 n n+1 u x1 ··· xn−1 ∈ J \ J , where we conclude that u x1 ··· xn−2 ∈ J \ J .

Again, there is α2 ∈ F \{0} where

2 n+1 n+1 u x1 ··· xn−2 + J = α2x1 ··· xn + J , which implies

3 n+2 n+2 u x1 ··· xn−2 + J = uα2x1 ··· xn + J .

3 n n+1 This leads to the fact u x1 ··· xn−3 ∈ J \ J . Continuing in the same manner, it

n n+1 n+1 can be shown that u + J = αnx1 ··· xn + J for some αn ∈ F \{0}. That is, J n = huni.

l l l+1 l l Now, assume J = hu i for some l ≥ n. Then J = hu1u , . . . , usu i. For each

l−1 l l−1 l j, uju ∈ J , so there is rj ∈ R such that uju = rju . Thus,

l l−1 l l+1 uju = (uju )u = (rju )u = rju .

It follows that J l+1 = hul+1i.

Lemma 2.5. Let R be a local ring with J = J(R) and F = R/J. Assume F is a prime field. Then R is semicommutative if and only if for any a, b ∈ J where ab = 0, aJb = 0.

Proof. Clearly, if R is semicommutative, the condition holds. Now, assume the condition holds. Let a, b ∈ R where ab = 0. If a or b is a unit then the other is 0 and

aRb = 0. So, assume a, b ∈ J. Let r ∈ R. Then r = r0 + r1 for some r0 ∈ F and

r1 ∈ J. Since F is prime,

arb = a(r0 + r1)b = ar0b + ar1b = ar0b = a(1 + ··· + 1)b = r0ab = 0. | {z } r0−times So, aRb = 0 and R is semicommutative. 45

Lemma 2.6. Let R be a local reflexive ring with J = J(R) and F = R/J. Assume that J 4 = 0 and F is a prime field. Let a, b, c ∈ J. Then abc = 0 if and only if bca = 0 if and only if cab = 0.

4 Proof. Let r ∈ R. Then r = r0 + r1 for some r0 ∈ F and r1 ∈ J. Since J = 0 and F is prime, ar(bc) = ar0(bc) = r0(abc). Therefore, abc = 0 if and only if aR(bc) = 0. Since the ring is reflexive, then abc = 0 if and only if (bc)Ra = 0, if and only if bca = 0, if and only if cab = 0.

In Lemma 2.6, the idea is that abc ∈ J(R)3 and the index of nilpotency of J(R)

K is at most 4. The result can be generalized to say that if a1 ··· an ∈ J where a1, . . . , an ∈ R and the index of nilpotency of J(R) is at most K + 1, then a1 ··· an = 0 if and only if ana1 ··· an−1 = 0.

2.3 Small Abelian Reflexive Rings

One of the main purposes of this chapter is to show that a minimal abelian reflexive non-semicommutative ring is of order 256 (which will be shown in the next section). This is a consequence of the primary result of this chapter, Theorem 2.12, which establishes that a local reflexive ring of order pk with k < 8 is semicommutative and therefore reversible. In particular, this implies that a local reflexive ring of an order less than 256 is semicommutative. To that end, the following five results are given which will ultimately be used to prove Theorem 2.12.

Proposition 2.7. Let R be a finite local ring, J = J(R) and F = R/J where F is prime. Assume J 3 = 0. Then R is semicommutative.

Proof. Let a, b ∈ J. Then aJb = 0. The result follows from Lemma 2.5.

Remark 2. In Theorem 3.2 of [6] it was shown that hu, vi R = F2 (u3, v2, vu, u2 − uv) 46

is a minimal non-reflexive semicommutative non-duo ring. Note that R is a finite local ring such that R/J(R) is prime and J(R)3 = 0. This shows that Proposition 2.7 cannot be strengthened to conclude the ring is reflexive and therefore reversible.

Proposition 2.8. Let R be a finite local ring, J = J(R) and F = R/J where F is

2 2 3 prime. Assume dimF (J/J ) ≤ 3 and dimF (J /J ) = 1. Then R is semicommutative.

Proof. Let K + 1 be the index of nilpotency of J. If J 3 = 0, by Proposition 2.7 R is

3 2 3 semicommutative. Assume J 6= 0. Then by Lemma 2.4, since dimF (J /J ) = 1

2 k k k+1 there exists u1 ∈ J \ J such that for k > 1, J = F u1 + J . Furthermore, since

2 2 dimF (J/J ) ≤ 3, J is at most 3 generated. So, J = F u1 + F u2 + F u3 + J for some u2, u3 ∈ J. PK j PK j Now, uiu1 = j=2 αju1 and u1ui = j=2 βju1 for some αj, βj ∈ F . Since F is a prime field, K−1 K−1 X j+1 X j+1 βju1 = u1uiu1 = αju1 j=2 j=2 K So, for 2 ≤ j ≤ K − 1, αj = βj showing uiu1 = u1ui + ti for some ti ∈ J . Now, PK j u2u3 = j=2 γju1 for some γj ∈ F . So, uiu2u3 = u2u3ui for any i. This shows that

given any 3-rd degree monomial in u1, u2 and u3, any permutation of the product is the same. By Lemma 2.5, it only needs to be shown that for any a, b ∈ J such that ab = 0, aJb = 0. Let a, r, b ∈ J such that ab = 0. By the property on the

permutation of triple products of u1, u2 and u3 and the fact that F is prime, arb = rab = 0. Hence, R is semicommutative.

Remark 3. The ring hu, vi R = F2 (u4, v2, vu, u3 − uv) 47

2 is a finite local ring such that R/J(R) is prime, dimF (J/J ) ≤ 3 and

2 3 dimF (J /J ) = 1. By Proposition 2.8, R is semicommutative. Since uv 6= 0 and vu = 0, it is non-reversible and therefore non-reflexive. This shows that Proposition 2.8 cannot be strengthened to include reflexivity in its conclusions.

Proposition 2.9. Let R be a finite local reflexive ring, J = J(R) and F = R/J

2 4 where F is prime. Assume dimF (J/J ) = 2 and J = 0. Then R is semicommutative.

Proof. By Lemma 2.5, it only needs to be shown that for any a, b ∈ J such that ab = 0, aJb = 0. Let a, r, b ∈ J such that ab = 0. If a, r or b is in J 2 then arb = 0 since J 4 = 0. So, assume a, r, b ∈ J \ J 2. If a + J 2 = γr + J 2 for γ ∈ F then arb = γr2b = rab = 0. So, assume J = ha, ri. Then b = αa + βr + t where α, β ∈ F and t ∈ J 2. If α = 0 then arb = βar2 = abr = 0. By Lemma 2.6, since rab = 0, if β = 0 then arb = αara = bra = rab = 0. Finally, assume α, β 6= 0. Then, since ab = 0,

0 = abr = αa2r + βar2

0 = a2b = αa3 + βa2r

0 = aba = αa3 + βara.

So, βar2 = −αa2r = −αara and so arb = αara + βar2 = 0. Hence, R is semicommutative.

Remark 4. In the previous proposition reflexivity was not superfluous as can be seen by noting the ring hu, vi F2 . (u2, v2, uvu − vuv) 48

This ring is a finite local non-semicommutative non-reflexive ring (see [6]) with a Jacobson radical that is 2-generated having index of nilpotency of 4. This example also shows the necessity of reflexivity in the following result.

Proposition 2.10. Let R be a finite local reflexive ring, J = J(R) and F = R/J

2 3 4 where F is prime. Assume dimF (J /J ) ≤ 2 and J = 0. Then R is semicommutative.

Proof. By Lemma 2.5, it only needs to be shown that for any a, b ∈ J such that ab = 0, aJb = 0. Note that Lemma 2.6 will be needed frequently throughout this proof so it will be used freely. Let a, r, b ∈ J such that ab = 0. If a, r or b is in J 2, then arb = 0 since J 4 = 0. So, assume a, r, b ∈ J \ J 2. If ar ∈ J 3, rb ∈ J 3 or ba ∈ J 3, then it can be shown that arb = 0. Assume, ar, rb, ba ∈ J 2 \ J 3. If ar + J 3 or rb + J 3 is a nonzero scalar multiple of ba + J 3, then arb = 0. So, assume neither

2 3 is the case. Then dimF (J /J ) = 2, and therefore there exists α, β ∈ F \{0} and γ ∈ F such that αar + βrb + γba ∈ J 3. If ra ∈ J 3 then 0 = ara = a2r showing βarb = αa2r + βarb + γaba = a(αar + βrb + γba) = 0. Since β 6= 0, arb = 0 in this case. So assume ra ∈ J 2 \ J 3. If ra + J 2 = ar + J 2, then arb = rab = 0. Assume J 2 = F ra + F ar + J 3. So, ba + J 2 = δ(ra + J 2) + ζ(ar + J 2) for δ, ζ ∈ F . Since the case when δ = 0 has already been considered, assume δ 6= 0. Then 0 = bab = δrab + ζarb = ζarb. If ζ 6= 0 then arb = 0. Finally, assume ζ = 0. Then ba + J 2 = δra + J 2. Then 0 = δ−1aba = ara showing 0 = (αar + βrb + γba)a = βrba = βarb. Since β 6= 0, arb = 0 in this case. Hence, R is semicommutative.

Proposition 2.11. Let R be a finite local reflexive ring, J = J(R) and F = R/J,

2 2 3 3 4 where F is prime. Assume dimF (J/J ) = 2, dimF (J /J ) = 2, dimF (J /J ) = 1, J 4 6= 0 and J 5 = 0. Then R is semicommutative. 49

Proof. By Lemma 2.4, J = F u + F v + J 2, J 2 = F u2 + F w + J 3, J 3 = F u3 + J 4 and J 4 = F u4 for u, v ∈ J \ J 2 and w ∈ {uv, vu, v2}. For some λ ∈ F , vu3 = λu4 and so (v − λu)u3 = 0. Replace v with v − λu. Since F is prime and J 5 = 0, vRu3 = 0. Since R is reflexive, u3Rv = 0 and u3v = 0. Using the reflexivity of R, it can be shown then that any 4-th degree monomial involving v is 0. Then any 3-rd degree monomial involving v is annihilated by both u and v on both the right and the left showing these monomials are in J 4. Given αu2 + βuv + γuv + δu2 ∈ J 3, u(αu2 + βuv + γuv + δu2) ∈ J 4 showing α = 0 since u3 ∈ J 3 \ J 4. This shows that uv + J 3, vu + J 3 and v2 + J 3 are each multiples of w + J 3. Now, uvu = αu4 for some α ∈ F . So, u(vu − αu3) = 0. By reflexivity, (vu − αu3)u = 0 and then vu2 = αu4 = uvu. Similarly, u2v = uvu so, u2v = uvu = vu2. Also, v2u = βu4 for some β ∈ F . So, (v2 − βu3)u = 0. By reflexivity, u(v2 − βu3) = 0 and then uv2 = βu4 = v2u. This shows that u commutes with any element of J 2. If vu ∈ J 3 or uv ∈ J 3 then vuv = 0 and by reflexivity, uv2 = v2u = vuv = 0. Alternatively, if it were assumed that v2 ∈ J 2 \ J 3, then w = v2. So, uv = δv2 + t for some δ ∈ F and t ∈ J 3 and then vuv = δv3 = uv2. Then by reflexivity again, vuv = uv2 = v2u. If instead u2v 6= 0, then vuv = γu2v for some γ ∈ F , so v2u = γvu2 = γu2v = vuv. Hence, v2u = vuv = uv2 in this case as well. Clearly, if v2 ∈ J 4, then v2u = vuv = uv2. In any of these cases, v commutes with any element of J 2. Next, it is shown that if none of these situations arise, the ring is not reflexive.

2 2 3 4 Assume uvu = 0, vuv 6= 0, J = hu , uvi, vu = α1uv + α2u + α3u and

2 3 4 v = γ1u + γ2u where α1 6= 0 and γ1 6= 0. Then

2 4 5 4 4 0 = uvu = α1u v + α2u + α3u = α2u , so α2 = 0. Then vu = α1uv + α3u and

2 2 2 2 vuv = α1uv = α1vuv. So, α1 = 1. If α1 = 1, then vuv = uv . Thus, in addition,

3 assume α1 6= 1. It will be shown in this case that R is not reflexive. Let a = uv + u 50

−1 and b = u − γ1 v. Let r ∈ R where r = r0 + r1 with r0 ∈ F and r1 ∈ J. Since aJ ⊂ J 4,

3 −1 arb = ar0b + ar1b = ar0b = (uv + u )r0(u − γ1 v)

4 −1 2 4 −1 4 4 4 = r0(u − γ1 uv ) = r0(u − γ1 γ1u ) = r0(u − u )

= 0

but

−1 3 4 −1 4 −1 4 ba = (u − γ1 v)(uv + u ) = (u − γ1 vuv)(u − γ1 α1γ1u )

4 4 4 = (u − α1u ) = (1 − α1)u

6= 0.

Since R is reflexive, this is not true. Therefore, in any case, u and v commute with the elements of J 2. So, any product of three elements from J is equal to any permutation of the product. Thus, by Lemma 2.6, R is semicommutative.

The five results in this section can be summarized in the following way. Let R be a finite local ring, J = J(R) and F = R/J where F is prime. If any of the following are true then R is semicommutative.

• J 3 = 0.

2 2 3 • dimF (J/J ) ≤ 3 and dimF (J /J ) = 1.

2 4 • R is reflexive, dimF (J/J ) = 2 and J = 0.

2 3 4 • R is reflexive, dimF (J /J ) ≤ 2 and J = 0.

2 2 3 3 4 4 • R is reflexive, dimF (J/J ) = 2, dimF (J /J ) = 2, dimF (J /J ) = 1, J 6= 0 and J 5 = 0. 51

Theorem 2.12. A finite abelian reflexive ring of order pk for some prime p and k < 8 is reversible.

Proof. Let R be a finite local reflexive non-semicommutative ring of order pl for some prime p, J = J(R), K + 1 be the index of nilpotency of J and F = R/J. Since R is finite and local, by Lemma 2.1, F is a field and |R| = |F |r for some r. Since R is non-semicommutative, it is non-symmetric. Then by Proposition 2.3, R is not a chain ring. Then r > 2. Now, r 6= 3, because otherwise since R is not a chain ring, J 2 = 0 meaning R is semicommutative which it is not. So, r ≥ 4. Then |R| ≥ |F |4. If |F | is not prime, l ≥ 8. So, assume |F | is prime. By Proposition 2.7, J 3 6= 0. By

2 2 3 i i+1 Proposition 2.8, if dimF (J/J ) ≤ 3 then dimF (J /J ) > 1. Let di = dimF (J /J ) and D = (d1, d2, . . . , dK ). If l < 8, then the only possibilities for D are (2, 2, 1), (2, 2, 2), (2, 3, 1), (3, 2, 1), (4, 1, 1) and (2, 2, 1, 1). By Proposition 2.9, D/∈ {(2, 2, 1), (2, 2, 2), (2, 3, 1)}. By Proposition 2.10, D/∈ {(3, 2, 1), (4, 1, 1)}. Finally, by Proposition 2.11, D 6= (2, 2, 1, 1). Hence, l ≥ 8. Since an abelian reflexive semicommutative ring is a direct sum of local reflexive semicommutative rings, any finite abelian reflexive ring of order pk for some prime p and k < 8 is semicommutative. Since reflexive semicommutative rings are reversible, the result follows.

2.4 Examples of Abelian Reflexive non-Semicommutative Rings

Let G be a group, written multiplicatively, and F a field. The group algebra FG is the set of all formal linear combinations of finitely many elements of G with coefficients in F . Thus an element of FG can be written as

X agg g∈G 52

where ag = 0 for all but finitely many elements of g ∈ G. Addition and multiplication in FG are defined as

X X X agg + bgg = (ag + bg)g g∈G g∈G g∈G and ! ! X X X agg bgg = (agbh)gh. g∈G g∈G g,h∈G Multiplication by scalars from F is naturally defined as ! X X α agg = (αag)g. g∈G g∈G From these definitions it follows that FG is an algebra. The identity is 1e, where e is the identity of G and 1 is the multiplicative identity of the field F . Moreover, FG is commutative if and only if G is commutative. The first example of a minimal abelian reflexive non-semicommutative ring is a

4 2 2 non-commutative group algebra. Let D8 = hr, s|r = s = (sr) = 1i be the Dihedral group of order 8.

Example 2.13. Let R = F2D8 and B = {1 + a|a ∈ D8 \{1}}. Then {1} ∪ B is an F -basis for the algebra R. It can be shown that hBi is an ideal of R and that hBi is a nilpotent ideal. Then any element outside hBi is a unit showing R is local. Next, notice (1 + rs)(r + s) = 0 but (1 + rs)s(r + s) 6= 0 so R is not semicommutative. It is now shown that R is reflexive. Let S be the subspace of R with basis {1, r, r2, r3} and define ¯ : S → S via

2 3 2 3 ¯ : α0 + α1r + α2r + α3r 7→ α0 + α3r + α2r + α1r .

This map is clearly an automorphism on S. Let

a = a1 + a2s, b = b1 + b2s ∈ S + Ss = R. Assume aRb = 0. So,

¯ ¯ 0 = ab = (a1 + a2s)(b1 + b2s) = (a1b1 + a2b2) + (a1b2 + a2b1)s 53 and

3 ¯ 3 ¯ 0 = arb = (a1 + a2s)r(b1 + b2s) = (a1rb1 + a2r b2) + (a1rb2 + a2r b1)s.

¯ 2 ¯ ¯ 2 Then a1b1 + a2b2 = 0 and a1b1 + a2r b2 = 0. So, a1b1 = a2b2 and a1b1 = r a1b1.

From the last result we can deduce that a1b1 = a1b1. So,a ¯2b2 = a1b1 and a1b1

2 annihilates 1 + r . Similarly, it can be shown thata ¯2b1 = a1b2 and a1b2 annihilates

2 1 + r . Using that asb = 0 and arsb = 0 it can be deduced thata ¯1b2 = a2b1,

2 a¯1b1 = a2b2, and both a2b1 and a2b2 annihilate 1 + r . Before proceeding note that

2 for y ∈ S, y +y ¯ ∈ h1 + r i. To finish, let x = x1 + x2s ∈ S + Ss = R. Then

bxa = (b1 + b2s)(x1 + x2s)(a1 + a2s)

= b1x1a1 + b1x2a¯2 + b2x¯1a¯2 + b2x¯2a1

+(b1x1a2 + b1x2a¯1 + b2x¯1a¯1 + b2x¯2a2)s

= b1a1(x1 +x ¯1) + b2a1(x2 +x ¯2)

+(b1a2(x1 +x ¯1) + b2a2(x2 +x ¯2))s

= 0.

Hence, R is a local reflexive non-semicommutative ring of order 256.

By Theorem 2.12, any abelian reflexive ring of order less than 256 is semicommutative. So, the group algebra F2D8 of Example 2.13 is a minimal abelian reflexive non-semicommutative ring providing the following theorem.

Theorem 2.14. A minimal abelian reflexive non-semicommutative ring has order

256 an example of which is F2D8.

Next, an example of a minimal abelian reflexive non-semicommutative ring where the index of nilpotency of the Jacobson radical is 4 is provided. One can see

2 2 3 3 4 that dimF (J/J ) = 3, dimF (J /J ) = 2 and dimF (J /J ) = 1. Therefore this is a 54

type of ring that is outside the rings described in Propositions 2.9 and 2.10. Furthermore, in Example 2.13, the nilpotency of the Jacobson radical is greater

3 than 4 (in F2D8, (1 + r), (1 + s) ∈ J(F2D8) but (1 + r) (1 + s) 6= 0) so, the two rings are non-isomorphic.

Example 2.15. Let

hu, v, wi R = F2 . ((u)2, (v)2, (w)2, uv, vw, wu, uwv + vuw, uwv + wvu)

Let x0 = 1, x1 = u, x2 = v, x3 = w, x4 = uw, x5 = vu, x6 = wv, x7 = uwv. Then R is an F2 algebra with basis {x0, . . . , x7}. Note that the ideal (u, v, w) is nilpotent and maximal. In addition, since x1, . . . , x7 ∈ (u, v, w) then for any x ∈ R, either x is a unit or 1 + x is a unit. Therefore R is local. Since uv = 0 but uwv 6= 0, R is non-semicommutative. We now show R is reflexive. Let a, b ∈ R and assume aRb = 0. Since ab = 0, if a or b is a unit, the other is 0 P7 P7 and bRa = 0. So, assume a, b ∈ J(R). Then a = i=1 aixi and b = i=1 bixi for some ai, bi ∈ F2. Since ab = 0, aub = 0, avb = 0 and awb = 0, it can be shown that

a1b2 = 0, a1b3 = 0, a2b1 = 0, a2b3 = 0, a3b1 = 0, a3b2 = 0 and

a1b6 + a4b2 + a2b4 + a5b3 + a3b5 + a6b1 = 0.

P7 Let r ∈ R with r = i=0 rixi. Then

bra = r0(b1a3uw + b2a1vu + b3a2wv) +

r0(b1a6 + b2a4 + b3a5 + b4a2 + b5a3 + b6a1)uwv +

(b1r3a2 + b3r2a1 + b2r1a3)uwv

= 0.

Hence, R is reflexive. 55

In the previous two examples the residue fields of the rings is F2, so it is a prime field. In the next two examples of minimal abelian reflexive non-semicommutative rings, the residue field is

2 F4 = F2[x]/hx + x + 1i, which is not a prime field. Let w = x + hx2 + x + 1i, so w2 = w + 1.

Example 2.16. Let σ be the Frobenius automorphism on F4, and [v; σ][u] R = F4 hv2 + u2, uvi Then R is a local reflexive non-semicommutative ring of order 256, as we will shown

2 next. An element of R has the form a0 + a1v + a2u + a3u , where ai ∈ F4. Hence, |R| = 256. In addition, observe that R is a local ring with J(R) = hv, ui and residue ∼ field R/J = F4. Consequently, R is an abelian ring because the only idempotents of R are 0,1. Notice (u + v)2 = 0 but (u + v)w(u + v) = u2 6= 0, so R is non-semicommutative. We now prove that R is reflexive. Let a, b ∈ R be such that aRb = 0. Clearly, bRa = 0 if one of the a, b is a unit in R. Thus, we assume that

2 2 a, b ∈ J, and let r ∈ R. Then a = a1v + a2u + a3u , b = b1v + b2u + b3u , and there

3 are r0 ∈ F4 and m ∈ J such that r = r0 + m. Since J = 0, we have arb = ar0b.

2 Thus, if r0 = 1, then ar0b = [a1σ(b1) + a2b2]u and, if r0 6= 1, then

2 ar0b = abr0 + a1σ(b1)u . The condition aRb = 0 implies that ab = 0 and ar0b = 0, where we obtain a1σ(b1) = 0, a2b2 = 0, and so 0 = σ(a1)b1. On the other hand, since

3 J = 0, we also have bra = br0a. If r0 = 1 we obtain

2 br0a = ba = [b1σ(a1) + b2a2]u = 0, and if r0 6= 1 we get

2 br0a = bar0 + b1va = b1σ(a1)u = 0.

Therefore, bra = br0a = 0 and so bRa = 0. Hence R is reflexive. 56

2 Example 2.17. Let S = F4[u]/hu i and θ the automorphism of S such that

2 2 θ(a0 + a1u) = a0 + a1wu. Let S[v; θ] T = hv2i Then we claim that R is a local reflexive non-semicommutative ring of order 256. ∼ Observe that T is a local ring with J(T ) = hu, vi and residue field R/J(T ) = F4. Consequently, T is an abelian ring because the only idempotents of T are 0,1. Notice (u + v)(u + wv) = 0 but (u + v)w(u + wv) = wuv 6= 0, so T is non-semicommutative. It remains to prove that T is reflexive. To this aim, we first derive some relations. Observe that any r ∈ T can be written in the form t = t0 + m0, where

3 m0 ∈ J, t0 = ξ0 + ψ0w ∈ F4, and ξ0, ψ0 ∈ F2. Since J(T ) = 0, atb = at0b and bta = bt0a. If a = a1u + (a2 + a3u)v and b = b1u + b2v + b3uv, then direct computations shows that

2 2 ab = (a1b2 + a2b1w)uv at0b = ξ0ab + ψ0w ab + ψ0a1b2uv (2.4.1)

2 2 ba = (b1a2 + b2a1w)uv bt0a = ξ0ba + ψ0w ba + ψ0b1a2uv (2.4.2)

Let a, b ∈ T such that aT b = 0. Clearly, bT a = 0 if one of the a, b is a unit in T .

Thus assume that a, b ∈ J(T ). Since ab = 0 and at0b = 0 for all r0 ∈ F4, it follows from (2.4.1) that a1b2 = 0 and a2b1 = 0. In consequence, in view of (2.4.2) we have ba = 0 and bt0a = 0. Therefore, T is reflexive.

The definitions of the rings R and T in Example 2.16 and Example 2.17 are quite similar. So, one may suspect that these two rings are isomorphic. To exclude this possibility, we analyze the structure of the group of units of those rings.

[v; σ][u] Theorem 2.18. Let R = F4 , U(R) be the group of units of R, and hv2 + u2, uvi 2 N = {1 + a1v + a2u + a3u }. Then

∗ 1. U(R) = NF4; 57

∗ 2. N ∩ F4 = {1};

3. N C U(R) and every element of N has order at most 4;

∗ 4. U(R) = N o F4;

5. For all x ∈ U(R), x12 = 1; and

6. a = (1 + v + u + u2)w ∈ U(R) has order 12.

2 ∗ Proof. (1) Let a = a0 + a1v + a2u + a3u ∈ U(R). Since a0 ∈ F4, then we can express a as follows:

−1 −1 −1 2 ∗ a = (1 + a1σ(a0) v + a2a0 u + a3a0 u )a0 ∈ NF4.

∗ (2) Clearly, N ∩ F4 = {1}. Hence, every element a ∈ U(R) can be uniquely

∗ expressed as a product of an element from N and an element from F4. (3) It is easy to prove that N is a subgroup of U(R). Note that the map

∗ 2 f : U(R) → F4 given by a0 + a1v + a2u + a3u 7→ a0 is a group homomorphism with

2 kernel N. Hence N C U(R). The product of a = 1 + a1v + a2u + a3u ∈ N and 2 b = 1 + b1v + b2u + b3u ∈ N is

2 ab = 1 + (a1 + a2)v + (a2 + b2)u + (a3 + b3 + a1σ(b1) + a2b2)u .

2 3 2 4 Thus, a = 1 + (a1 + a2b2)u , whence a = 1. (4) follows from (1)-(3).

∗ (5) Every element in N has order at most 4 and every element of F4 has order at most 3. Then every element of U(R) has order at most 12 as 3 and 4 are relative prime. (6) It is easy to verify that 1 + v + u + u2 has order 4. Since w has order 3, then a = (1 + v + u + u2)w has order 12. 58

The structure of the group N in the above proposition is described in the next result.

2 Lemma 2.19. Let N = {1 + a1v + a2u + a3u }. Then

∼ N = (Z4 ⊕ Z4 ⊕ Z2) o Z2.

Proof. Let a = 1 + u, b = 1 + wu and x = 1 + v. Then a and b have order 4. Moreover, hai ∩ hbi = {1}, so haihbi is an abelian subgroup of N of order 16

2 2 isomorphic to Z4 ⊕ Z4. Note that ax = xa, bx = xb and x = a . Thus, the subgroup G of N generated by a, b, x is an abelian subgroup of N order 32. Hence ∼ G is a normal subgroup of N, and G = Z4 ⊕ Z4 ⊕ Z2. Direct computations show that every element in G has the form

2 1 + ξ1v + ξ2u + ξ3u ,

2 where ξ1 ∈ {0, 1}, and ξ2, ξ3 ∈ F4. Let y = 1 + wv + u. Then y 6∈ G and y = 1. It follows that G ∩ hyi = {1} and so N = Ghyi. Therefore, N = G o hyi, from where the result follows.

S[v; θ] Theorem 2.20. Let T = as described in Example 2.17, U(T ) be the group of hv2i units of T , and N = {1 + bv : b ∈ S} ⊆ U(T ). Then

1. N C U(T ) and every element of N has order at most 2.

∗ 2. N ∩ F4 = {1}.

3. U(T ) = N · U(S), where U(S) is the group of units of S.

∼ 2 4. U(T ) = N o U(S) = Z4 o (Z3 × Z2)

5. For all x ∈ U(T ), x6 = 1. 59

Proof. (1) Let 1 + bv, 1 + b0v ∈ N. Since (1 + bv)(1 + b0v) = 1 + (b + b0)v, it follows that N is a subgroup of U(T ) whose elements have order at most 2. Let

−1 −1 −1 a = a0 + a1v ∈ U(T ), then a = a0 + a0 θ(a0)a1v. Thus, for any 1 + bv ∈ U(T ) we have

−1 −1 −1 a(1 + bv)a = (a0 + a1v)(1 + bv)(a0 + a0 θ(a0)a1v)

−1 −1 = (a0 + (a0b + a1)v)(a0 + a0 θ(a0)a1v)

−1 −1 = 1 + (θ(a0)a1 + a0bθ(a0 ) + a1θ(a0 ))v ∈ N.

Therefore N C U(T ). ∗ (2) Clearly N ∩ F4 = {1}.

(3) Let a = a0 + a1v ∈ U(T ). Then a0 is a unit in S, so

−1 a = a0 + a1v = a0 + a1θ(a0) θ(a0)v

−1 −1 = a0 + a1θ(a0) va0 = (1 + a1θ(a0) v)a0 ∈ N · U(S).

(4) From (1) to (3) we have that U(T ) is the semidirect product of N and

∼ 4 U(S). Since |N| = |S| = 16 and every element in N has order at most 2, N = Z2.

∗ On the other hand, ξ + ψu ∈ U(S) implies that ξ ∈ F4 and ψ ∈ F4. Therefore,

ξ + ψu = ξ(1 + ξ−1ψu).

2 Observe that |{1 + ψu : ψ ∈ F4}| = 4 and that (1 + ψu) = 1, so ∼ 2 {1 + ψu : ψ ∈ F4} = Z2 is a subgroup of U(S). Since U(S) is abelian and ∗ ∼ 2 F4 ∩ {1 + ψu : ψ ∈ F4} = {1}, we have U(S) = Z3 × Z2. In particular, this implies that every unit in U(S) has order at most 6.

(5) Let x = (1 + bv)a ∈ U(T ), where b ∈ S and a = ξ0 + ξ1u ∈ U(S). Observe that

x2 = [(1 + bv)a] [(1 + bv)a] = (1 + bv) a(1 + bv)a−1 a2

= 1 + b(1 + aθ(a)−1)v a2. (2.4.3) 60

Thus, an inductive argument shows that

x6 = 1 + b 1 + aθ(a)−1 + a2θ(a)−2 + ··· + a5θ(a)−5
v

= 1 + b 1 + aθ(a)−1 + (aθ(a)−1)2 + ··· + (aθ(a)−1)5
v (2.4.4)

In the commutative ring S[z], the polynomial z6 − 1 factors as

z6 − 1 = (z − 1)(z5 + z4 + ··· + z + 1).

Moreover, if z − 1 ∈ S[z] is a unit, then

z6 − 1 = z5 + z4 + ··· + z + 1. (2.4.5) z − 1

−1 2 4 Let z = aθ(a) ∈ U(S). If ξ0 = 1, then a = 1 and so by (2.4.3) we get that x = 1.

If ξ0 6= 1, then ξ0 ∈ {w, w + 1}, so z − 1 ∈ U(S), where it follows from (2.4.5) that 1 + z + z2 + ··· + z5 = 0. Therefore x6 = 1.

[v; σ][u] Corollary 2.21. The ring R = F4 in Example 2.16 and the ring hv2 + u2, uvi S[v; θ] T = in Example 2.17 are non-isomorphic. hv2i

Proof. The ring R has units of order 12. For example, a = w + v + u is a unit in R of order 12 in R. On the other hand, all units in T have order at most 6. Hence R and T are non-isomorphic. 61 3 Morita Context Rings

3.1 Background on Morita Context Rings and Goals

A Morita context is a 6-tuple M = (A, M, N, B, φ, ψ) where A and B are rings,

M is an (A, B)-bimodule, N is a (B,A)-bimodule, and φ : M ⊗B N → A and   AM   ψ : N ⊗A M → B are bimodule homomorphisms such that T (M) =   is an NB associative ring with the obvious matrix operations. The ring T (M) is the Morita context ring associated with M. The ideals φ(M ⊗ N) ⊆ A and ψ(N ⊗ M) ⊆ B are called the trace ideals of T (M). In the special case when φ and ψ are zero, T (M) is called a Morita context ring with zero trace ideals. If N = 0 then T (M) is called a formal triangular matrix ring. For more on Morita context rings see [83–87]. Historically, Morita context rings were introduced by Bass in his exposition of the Morita Theorems [84]. Since then, Morita context rings (especially, formal triangular matrix rings) have been extensively used as a source to provide interesting examples in ring theory. This gives rise to the problem of characterizing Morita context rings that possess a certain ring property. This problem has been addressed in the literature for some families of rings (see, e.g., [10, 11, 87]). For instance, characterizations of prime and semiprime rings in the framework of Morita context rings are given by Nicholson and Watters [10] and Hongan [11], respectively. Also, Proposition 1.35 establishes that the formal upper triangular matrix ring   AM     is 2-primal if and only if A and B are 2-primal rings. Also, by 0 B   AM   Proposition 1.39, the formal upper triangular matrix ring   is NR if and 0 B only if A and B are NR. 62

The aim in this chapter is to characterize Morita context rings which are reflexive, 2-primal, weakly 2-primal and NI. The concept of faithful bimodule homomorphism is introduced in order to simplify characterizations of reflexive Morita context rings. For a Morita context M = (A, M, N, B, φ, ψ), we prove that T (M) is reflexive if and only if A and B are reflexive, φ or ψ are faithful (bimodule homomorphisms), and for all m ∈ M and n ∈ N, mBn = 0 if and only if nAm = 0. We show how this result is connected to the characterizations of prime and semiprime Morita context rings given in [10] and [11], respectively. Also, we establish the connection between reflexive Morita context rings, and torsionless and faithful modules. On the other hand, we show that a Morita context ring T (M) is NI if and only   N(A) M   if A and B are NI rings and N(T (M)) =  , where N(R) denotes NN(B) the set of nilpotent elements of a ring R. In order to simplify this result, we posed the question whether a Morita context ring T (M) is NI if and only if A and B are NI rings, and φ(M ⊗ N) ⊆ N(A) and ψ(N ⊗ M) ⊆ N(B). We show that this is the case if and only if K¨othe’sconjecture holds. Later, we show that it is true in any case, however, that a Morita context ring T (M) is weakly 2-primal (2-primal) if and only if A and B are weakly 2-primal (2-primal) rings, and φ(M ⊗ N) ⊆ N(A) and ψ(N ⊗ M) ⊆ N(B). Throughout this chapter all modules are assumed to be unitary, unless otherwise stated. Given a Morita context M = (A, M, N, B, φ, ψ), note that if

B = 0 then T (M) ∼= A . Similarly, if A = 0 then T (M) ∼= B. Thus, for the purposes of this dissertation, we assume that A and B are non-zero rings. In addition, for the sake of notational convenience, for any a ∈ A and b ∈ B, define the 63     a 0 0 0     formal matrices aE1 =   and bE2 =   in T (M). In particular, as 0 0 0 b usual E1 and E2 denote the idempotents 1E1 and 1E2, respectively. For simplicity, we also write φ and ψ multiplicatively, so that mn := φ(m ⊗ n), nm := ψ(n ⊗ m), MN = φ(M ⊗ N) and NM = ψ(N ⊗ M).

3.2 Reflexive Morita Context Rings

Recall that a ring R is called reflexive if aRb = 0 implies bRa = 0 for all a, b ∈ R. Recall that a ring R is semiprime if aRa = 0 implies a = 0 for all a ∈ R, while R is prime if aRb = 0 implies a = 0 or b = 0 for all a, b ∈ R. It is clear from the definitions that prime semiprime reflexive. It is known that reflexive, semiprime and prime are Morita invariant ring properties (c.f. [45, Theorem 2.6] and [88]). Furthermore, it is straightforward to verify that these ring properties pass from a ring R to eRe for any idempotent e ∈ R. This result applied to a Morita context ring yields the next result.

Lemma 3.1. Let M = (A, M, N, B, φ, ψ) be a Morita context. If T (M) is reflexive (resp., semiprime, prime) then A and B are reflexive (resp., semiprime, prime).

∼ ∼ Proof. Clearly, A = E1T (M)E1 and B = E2T (M)E2.

Given a ring R and a left R-module RM, there is an isomorphism of left

R-modules f : R ⊗R M → M given by r ⊗ m 7→ rm. This map satisfies the property: “for all m ∈ M, if f(R ⊗ m) = 0, then m = 0”. Although the property

“for all r ∈ R, if f(r ⊗ M) = 0, then r = 0” is satisfied if and only if RM is a faithful module. Motivated by these two properties, and the fact that R and RM are both (R, Z)-bimodules, we introduce the concept of faithful bimodule 64

homomorphism, which will be essential in all characterizations of reflexive, semiprime and prime Morita context rings.

Definition 3.2. Let A, B, C rings, and AXB, BYC and AZC bimodules. An

(A, C)-bimodule homomorphism f : X ⊗B Y → Z is called faithful if both of the following conditions are satisfied:

1. for all x ∈ X, if f(x ⊗ Y ) = 0, then x = 0;

2. for all y ∈ Y , if f(X ⊗ y) = 0 then y = 0.

In the notation of the previous definition, note that the zero bimodule homomorphism is faithful provided that both bimodules X and Y are zero. Clearly, this fact is independent of the bimodule Z. Thus a faithful bimodule homomorphism need not be onto. Furthermore, it is not difficult to see that the faithfulness property of a bimodule homomorphism is independent from its one-to-one and isomorphism attributes.

3.2.1 Reflexive Morita Context Rings

We start introducing a condition related to reflexivity on Morita contexts, which is needed in the characterization of reflexive Morita context rings.

Definition 3.3. A Morita context M = (A, M, N, B, φ, ψ) is called reflexive if mBn = 0 if and only if nAm = 0 for all m ∈ M and n ∈ N.

Remark 5. Given a Morita context M, the terminology “M is reflexive” refers to the reflexivity condition introduced in Definition 3.3, not to be confused with the notion of “reflexive Morita context ring”, which means that T (M) is a reflexive ring.

Lemma 3.4. Let M = (A, , M, N, B, φ, ψ) be a Morita context and a ∈ A, b ∈ B, m, µ ∈ M and n, ν ∈ N. If A is reflexive and φ is faithful, then 65

1. aAm = 0 if and only if mNa = 0,

2. nAa = 0 if and only if aMn = 0. If B is reflexive and ψ is faithful, then 3. bBn = 0 if and only if nMb = 0,

4. mBb = 0 if and only if bNm = 0. If A is reflexive and φ is faithful, or, B is reflexive and ψ is faithful, then 5. mNµ = 0 if and only if µNm = 0,

6. nMν = 0 if and only if νMn = 0,

7. aMb = 0 if and only if bNa = 0.

Proof. We only prove 1) and 5) since the other statements can be proved similarly. To show 1), suppose first that aAm = 0. Then aA(mN) = 0. Since A is reflexive, (mN)Aa = 0. The ring A has identity, so mNa = 0. Conversely, if mNa = 0, then (mN)Aa = m(NA)a = 0. Because A is reflexive, (aAm)N = aA(mN) = 0, where we conclude that aAm = 0 because φ is faithful. To see 5), suppose without loss of generality that A is reflexive and φ is faithful. If mNµ = 0 then (mN)Aµ = m(NA)µ = 0, so (mN)A(µN) = 0. The ring A has identity and is reflexive, so (µNm)N = (µNAm)N = (µN)A(mN) = 0. Therefore µNm = 0 because φ is faithful. Clearly, the reverse implication can be shown in a similar fashion.

Reflexive Morita context rings are characterized in the next result.

Theorem 3.5. Let M = (A, M, N, B, φ, ψ) be a Morita context. The following are equivalent. 1. T (M) is reflexive.

2. A, B are reflexive rings, φ or ψ is faithful, and M is reflexive. 66

3. A, B are reflexive rings, φ and ψ are faithful, and M is reflexive.

Proof. 1) ⇒ 2) By Lemma 3.1, A and B are reflexive. If φ is not faithful, then one of the conditions in Definition 3.2 does not hold for φ. Without loss of generality, suppose that the condition “for all x ∈ X, f(x ⊗ Y ) = 0 ⇒ x = 0” is false, and let   0 m0 a   a =   ∈ T (M), where m0 ∈ M \{0} and m0N = 0. Then T (M)E1 = 0 0 0 but E1T (M)a 6= 0 (a contradiction). Hence φ is faithful. Similarly we show that ψ is faithful. Finally, M is reflexive because T (M) is reflexive and for all m ∈ M and       0 m 0 0 mBn 0       n ∈ N,   T (M)   =   and 0 0 n 0 0 0       0 0 0 m 0 0         T (M)   =  . n 0 0 0 0 nAm 2) ⇒ 3) Without loss of generality assume that φ is faithful. Let n ∈ N be such that nM = 0. Then nAM = 0, and because M is reflexive, Mn = MBn = 0. Since φ is faithful, n = 0. Similarly, we prove that Nm = 0 implies m = 0. Therefore ψ is faithful.     a m α µ a   b   a b 3) ⇒ 1) Let =   , =   ∈ T (M) such that T (M) = 0. We n b ν β want to prove that bT (M)a = 0, or equivalently, if we expand the product bT (M)a, we want to show that

αAa + µNa + αMn + µBn = 0, (3.2.1)

αAm + µNm + αMb + µBb = 0, (3.2.2)

νAa + βNa + νMn + βBn = 0, (3.2.3)

νAm + βNm + νMb + βBb = 0. (3.2.4)

To this aim, we prove that every summand involved in each of the relations (3.2.1)-(3.2.4) is zero. In fact, from aT (M)b = 0 we deduce that 67 a(EiT (M)Ej)b = 0 for all 1 ≤ i, j ≤ 2. Thus a(E1T (M)E1)b = 0, where we obtain aAα = 0, aAµ = 0, nAα = 0, and nAµ = 0. Because A is reflexive, aAα = 0 implies αAa = 0. In addition, since φ is faithful, by Lemma 3.4, the relations aAµ = 0 and nAα = 0 imply µNa = 0 and αMn=0. Moreover, by hypothesis M is reflexive, so nAµ = 0 implies µBn = 0. Thus αAa + µNa + αMn + µBn = 0. Similarly, using the relations a(E2T (M)E1)b = 0, a(E1T (M)E2)b = 0 and a(E2T (M)E2)b = 0 along with Lemma 3.4 we prove that (3.2.2), (3.2.3) and (3.2.4) hold, respectively. Therefore T (M) is reflexive.

In [45], Kwak and Lee proved that the ring Un(R) of n × n upper triangular matrices over any ring R is non-reflexive. In light of Theorem 3.5 we now can extend that result to formal triangular matrix rings. Moreover, note that Theorem 3.5 gives simple ways to create non-reflexive rings via Morita context ring constructions.   AM   Corollary 3.6. The formal triangular matrix ring   is non-reflexive 0 B whenever M 6= 0 or one of the corner rings A or B is non-reflexive.

Let M = (A, M, N, B, φ, ψ) be a Morita context such that A is reflexive and φ is faithful, or B is reflexive and ψ is faithful. Then part 7) of Lemma 3.4 establishes that the modules M and N have the same annihilators in A and B; that is,

AnnA(M) = AnnA(N) and AnnB(M) = AnnB(N). This remark proves the following corollary of Theorem 3.5.

Corollary 3.7. If M = (A, M, N, B, φ, ψ) is a Morita context such that T (M) is reflexive. Then AnnA(M) = AnnA(N) and AnnB(M) = AnnB(N). In particular,

AM is faithful if and only if NA is faithful; and MB is faithful if and only if BN is faithful. 68

We now investigate how faithfulness of the bimodules M and N in a Morita context M = (A, M, N, B, φ, ψ) imply the property of M being reflexive.

Proposition 3.8. Let M = (A, M, N, B, φ, ψ) be a Morita context. Assume that one of the following holds:

1. A is reflexive, φ is faithful, ψ is onto and BN is faithful.

2. B is reflexive, ψ is faithful, φ is onto and AM is faithful. Then T (M) is a reflexive ring.

Proof. Assume that 1) holds. To see that T (M) is a reflexive ring, by Theorem 3.5, we only need to prove that B is reflexive and M is reflexive. Suppose that b, β ∈ B are such that bBβ = 0. Since φ is onto and A has identity, NAM = B. Thus bNAMβ = 0, which implies (MbN)A(MβN) = 0. The ring A is reflexive, so (MβN)A(MbN) = 0. Note that βNAMbN ⊆ N. Hence using that φ is faithful we get that βNAMbN = 0, whence βNAMb = 0 because BN is faithful. Since NAM = B we obtain βBb = 0, and so B is reflexive. To prove that M is reflexive, let m ∈ M and n ∈ N. If mBn = 0 then mNAMn = 0, where we obtain MnAmN = 0 because A is reflexive. Since φ is faithful, MnAmN = 0 implies nAmN = 0. The module BN is faithful, so nAm = 0. Conversely, if nAm = 0 then MnAmN = 0 and so 0 = mNAMn = mBn because A is reflexive and ψ is onto. Therefore, M is reflexive. Similarly we show that T (M) is reflexive provided that 2) holds.

Proposition 3.9. Let M = (A, M, N, B, φ, ψ) be a Morita context, where A and B

are reflexive rings and φ and ψ are faithful. Also assume that AMB or BNA is faithful. Then T (M) is a reflexive ring.

Proof. Since A is reflexive and φ is faithful, we conclude that AMB and BNA are faithful. Now, assume that nAm 6= 0. Then MnAmN 6= 0 because φ and ψ are 69 faithful, and AMB and BNA are faithful bimodules. Since A is reflexive, mNAMn 6= 0, and so mBn 6= 0 because NAM ⊆ B. Similarly we prove that mBn 6= 0 implies nAm 6= 0. The result follows by Theorem 3.5.

We close this section on reflexive Morita context rings with some specific examples that partially motivated this dissertation.

Example 3.10. For any ring R and any integer n ≥ 1, let Rn×1 be the set of all columns with n entries from R. By Corollary 3.6, the Morita context ring   M (R) Rn×1  n   , which is the ring of (n + 1) × (n + 1) upper triangular matrices 0 R over R, is non-reflexive. The particular case when n = 2 and R = F2, the finite field with 2 elements, was addressed in [5, Theorem 3.7].

Example 3.11. Let F be a field and A = F [x]/(xn). Note that the commutative ring A is local with maximal ideal (x) and residue field A/(x) ∼= F . Thus the additive group (F, +) can be viewed as an (A, A)-bimodule in a natural way. Let φ : F ⊗ F → F [x]/(xn) and ψ : F ⊗ F → F [x]/(xn) be given by the formulas

φ : m ⊗ n 7→ mnxn−1 and ψ : n ⊗ m 7→ nmxn−1.

Then φ and ψ are bimodule homomorphisms which satisfy the associative relations n1(mn2) = (n2)m and m1(nm2) = (m1n)m2 for all mi, m2 ∈ F and n1, n2 ∈ F . These associative relations imply that M = (A, F, F, A, φ, ψ) is a Morita context (see [83, pp. 273]). Moreover, it is easy to see that φ and ψ are faithful, and M is   F [x]/(xn) F   reflexive. Hence T (M) =   is a reflexive ring in view of FF [x]/(xn)   2 F2[x]/(x ) F2 Theorem 3.5. In particular, the finite ring R =   of order  2  F2 F2[x]/(x ) 70

64 is reflexive. This case has been noticed in [5, Theorem 3.7], where it is also proved that R is a reflexive non-NI ring of minimal order.

Note that in the previous example neither φ nor ψ is onto. Also, observe that neither M nor N is a faithful bimodule. Thus Example 3.11 illustrates that the conditions in Propositions 3.8 and 3.9 are not always satisfied by a reflexive Morita context ring.

3.2.2 Semiprime Morita Context Rings

In [11], Hongan characterizes semiprime Morita context rings. Since semiprime rings are reflexive, Theorem 3.5 gives insights of Hongan’s result. For completeness and further reference, we include this characterization and add some remarks.

Theorem 3.12 ([11, Proposition 1]). Let M = (A, M, N, B, φ, ψ) be a Morita context. The following are equivalent.

1. T (M) is semiprime.

2. A and B are semiprime, and φ or ψ is faithful.

3. A and B are semiprime, and φ and ψ are faithful.

It is worth noting that conditions2) and3) in Theorem 3.12 cannot be weakened; that is, both rings A and B must be assumed to be semiprime and at least one of the the maps φ or ψ must be faithful. In fact, by Theorem 3.5, any   AM   Morita context ring T (M) =   with zero trace ideals and one non-zero NB module M or N is non-reflexive, and so non-semiprime regardless whether A and B are semiprime or not. Moreover, the next example shows that the condition “A is semiprime, φ and ψ are faithful” does not imply that B is semiprime. 71

Example 3.13. Let A = F be a field and B = F [x]/(x2). Then the Morita context   A 0   ∼ ring T =   is not semiprime because T = A ⊕ B. In this case, φ and ψ are 0 B the zero bimodule homomorphisms, whence faithful because the modules M and N are zero.

Since semiprime rings are reflexive, it is reasonable to expect that, in the same manner that faithful bimodules are related to reflexive Morita context rings, faithful bimodules are related to semiprime Morita context rings. We explore these relations in the next result.

Proposition 3.14. Let M = (A, M, N, B, φ, ψ) be a Morita context. Consider the following conditions.

1. A is semiprime, φ is faithful, and MB or BN is faithful.

2. A is semiprime, φ or ψ is faithful, and for all b ∈ B, if MbN = 0 then b = 0.

3. B is semiprime, ψ is faithful, and AM or NA is faithful.

4. B is semiprime, φ or ψ is faithful, and for all a ∈ A, if NaM = 0 then a = 0.

Then (1) is equivalent to (2); (3) is equivalent to (4); and if any of these conditions holds then T (M) is semiprime.

Proof. 1) ⇒ 2) Let b ∈ B and assume that MbN = 0. Since φ is faithful, we get

Mb = 0 and bN = 0. Hence b = 0 because MB or BN is faithful.

2) ⇒ 1) We first observe that BN is faithful because if bN = 0, then MbN = 0 and so b = 0. The conclusion is complete if φ is faithful. Thus, assume that ψ is faithful and let m ∈ M such that mN = 0. If m 6= 0 then Nm 6= 0 because ψ is faithful. Hence there exists n ∈ N such that nm 6= 0. Let b = nm. Then b 6= 0. On the other 72

hand, notice that MbN = 0, so b = 0, which is a contradiction. Thus m = 0. Likewise we show that Nm = 0 implies m = 0. Therefore φ is faithful. The fact that3) is equivalent to4) is proven in a similar way. Finally, we show that any of these condition implies that T (M) is semiprime. Assume1) holds. By Theorem 3.12, we only need to show that B is semiprime. Let b ∈ B and assume that bBb = 0. Then b(NAM)b = 0, so (MbN)A(MbN) = 0.

Since A is semiprime, MbN = 0 and so b = 0 because φ is faithful and MB or BN is faithful. Likewise it is shown that3) implies that T (M) is semiprime.

It is natural to ask whether conditions1) and3) in Proposition 3.14 are equivalent. The following example erases this possibility. It also shows that none of those conditions is necessary.

Example 3.15. Let R be a semiprime ring and k, ` ≥ 1 be integers. Consider the rings A = Rk and B = R`, and let M = R and N = R be the additive abelian group

of the ring R. These groups become bimodules if for all a = (a1, . . . , ak) ∈ A,

b = (b1, . . . , b`) ∈ B, m ∈ R, and n ∈ N, we define am = a1m, mb = mb1,

bn = b1n,na = na1. Let φ : M ⊗ N → A and ψ : N ⊗ M → B be given by the formulas:

φ :(m ⊗ n) = (mn, 0,..., 0) and ψ(n ⊗ m) = (nm, 0,..., 0).

Then φ and ψ are bimodule homomorphisms which satisfy the associative relations:

n1(mn2) = (n2)m and m1(nm2) = (m1n)m2 for all mi, m2 ∈ M and n1, n2 ∈ N.

k ` Then Mk,` = (A, M, N, B, φ, ψ) is a Morita context. Since R and R are semiprime rings and ψ and φ are faithful, by Theorem 3.12, the next Morita context ring is 73 semiprime:     AM Rk R     T (Mk,`) =   =   NB RR`

Notice that for k ≥ 2, Mk,1 is a Morita context that satisfies condition1) but not3) in Proposition 3.14. On the other hand, for all ` ≥ 2, T (M1,`) satisfies condition3) but not1) in the same proposition. Moreover, for all k, ` ≥ 2, Mk,` does not satisfy conditions1) and3) in Proposition 3.14.

3.2.3 Prime Morita Context Rings

In Proposition 3.14 we give sufficient conditions for a Morita context ring to be semiprime. Without further assumptions, none of those conditions are necessary. We will show in the next result that the corresponding conditions yield characterizations of prime Morita context rings. In particular, we will prove that a Morita context ring T (M), where M = (A, M, N, B, φ, ψ), is prime if and only if A is prime, φ is faithful, and for all b ∈ B, MbN = 0 implies b = 0. This characterization of prime Morita context rings was obtained by Nicholson and Watters [10].

Theorem 3.16. Let M = (A, M, N, B, φ, ψ) be a Morita context. The following are equivalent.

1. T (M) is prime.

2. A and B are prime, φ or ψ is faithful, and M or N is non-zero.

3. A is prime, φ is faithful, and MB or BN is faithful.

4. A is prime, φ or ψ is faithful, and for all b ∈ B, if MbN = 0, then b = 0.

5. B is prime, ψ is faithful, and AM or NA is faithful. 74

6. B is prime, φ or ψ is faithful, and for all a ∈ A, if NaM = 0, then a = 0.

7. A and B are prime, φ and ψ are faithful, and AMB and BNA are faithful.

Proof. For notational convenience, let T = T (M). 1 ⇒ 2) By Lemma 3.1, A and B are prime. Moreover, since prime rings are semiprime rings, φ or ψ is faithful in light of Theorem 3.12. Now, if M and N are

zero then T (M) ∼= A ⊕ B, which is not a prime ring. Hence at least one of M or N is non-zero. But since φ is faithful, M is non-zero if and only if N is non-zero. 2) ⇒ 3) Since prime rings are semiprime, then Theorem 3.12 implies that φ and ψ are faithful. Thus, to conclude3) we only need to show that MB is faithful. Assume that Mb = 0. Then MBb = 0. Since M 6= 0 and ψ is faithful, N is also non-zero. Thus there are non-zero elements m ∈ M and n ∈ N such that nm 6= 0. Then nmBb = 0, which implies that b = 0. Hence MB is faithful. 3) ⇒ 4) Assume MbN = 0 for some b ∈ B. Since φ is faithful, Mb = 0 and bN = 0. Hence b = 0 because MB or BN is faithful. 4) ⇒ 5) To show that B is prime, assume that b, β ∈ B satisfy bBβ = 0. Then (MbN)A(MβN) = 0, where it follows that b = 0 or β = 0. Now, from Proposition

3.14 and Theorem 3.12 we obtain that ψ is faithful. Finally, if a ∈ AnnA(M) then a(AM)N = aMN = 0. But MN = M1N 6= 0 and so a = 0 because A is prime.

Therefore AM is faithful. 5) ⇒ 6) If NaM = 0 for some a ∈ A, then Na = 0 and aM = 0 because ψ is faithful. Hence a = 0. 6) ⇒ 7) Argue as in the proof of4) ⇒ 5) to prove that A is prime. From Proposition 3.14 and Theorem 3.12, we deduce that φ and ψ are faithful. Furthermore, if a ∈ Ann(M) then NaM = 0 which implies that a = 0. Also, if 75

b ∈ Ann(M) then (NM)Bb = 0 and so b = 0 because NM 6= 0 and B is prime. Thus M is faithful. By Corollary 3.7, N is faithful. A B   B A  a m α µ a   b   a b 7) ⇒ 1) Let =   ∈ T and =   ∈ T \{0} be such that T = 0. n b ν β

Then a(EiTEj)b = 0 for all 1 ≤ i, j ≤ 2. From these products we derive the following relations:

aAα = 0 mNα = 0 nAα = 0 bNα = 0 (3.2.5)

aAµ = 0 mNµ = 0 nAµ = 0 bNµ = 0 (3.2.6)

aMν = 0 mBν = 0 nMν = 0 bBν = 0 (3.2.7)

aMβ = 0 mBβ = 0 nMβ = 0 bBβ = 0 (3.2.8)

If α 6= 0 then we use the relation in (3.2.5) to prove that a = 0. Indeed, since α 6= 0 and A is prime, aAα = 0 implies a = 0. Also, mNα = 0 implies that (mN)Aα = 0, and so m = 0 because A is prime and φ is faithful. Similarly, the relation nAα = 0 implies n = 0. Now, from bNα = 0 we obtain (MbN)Aα = 0. This implies b = 0

because A is prime, φ is faithful, and BNA is faithful. Using similar arguments and the relations (3.2.6), (3.2.7), or (3.2.8) we prove that a = 0 if µ 6= 0, ν 6= 0 or β 6= 0, respectively.

A (left or right) R-module M is called prime if M 6= 0 and

AnnR(N) = AnnR(M) for every non-zero submodule N ≤ M. We say that a

bimodule AMB is prime if both AM and MB are prime modules.

Corollary 3.17. Let M = (A, M, N, B, φ, ψ) be a Morita context. If T (M) is prime then M and N are (necessarily faithful) prime bimodules.

Proof. Let L ≤ AM and K ≤ MB be a non-zero submodules as indicated. Let

a ∈ AnnA(L) and b ∈ AnnB(K). Then for any l ∈ L \{0} and k ∈ K \{0} we have 76

aAl = 0 and kBb = 0. Since φ and ψ are faithful, we obtain lN 6= 0 and Nk 6= 0. Hence aA(lN) = 0 and (Nk)Bb = 0 imply a = 0 and b = 0 because A and B are prime rings. Therefore AnnA(L) = 0 and AnnB(K) = 0. By Theorem 3.16, AMB is faithful, whence AMB is a prime bimodule module. Similarly we show that ANB is a faithful prime module.

According to Theorem 3.16 and Corollary 3.7, a Morita context ring T (M) is

prime if and only if A is prime, φ is faithful, and AMB is faithful. It is natural to ask

whether “AMB is faithful” can replaced by “AMB is prime” in that characterization. The answer to this question is in the negative as the following example shows.

Example 3.18. Let M1,2 = (A, M, N, B, φ, ψ) be the Morita context of Example

3.15 when the semiprime ring R is Z. The Morita context ring     AM   ZZ  T (M1,2) =   =   NB ZZ2 was shown to be a semiprime ring in Example 3.15, and so φ is faithful. Notice that in this case A is prime and M is a prime bimodule. However, by Theorem 3.16,

T (M1,2) is not a prime ring because B is not a prime ring.

3.3 NI Morita Context Rings

The main aim of this section is to characterize Morita context rings that are NI, weakly 2-primal or 2-primal. To this end, it is important to note that these three classes of rings are closed under subrings, possibly without identity (see Propositions 1.35, 1.36, and 1.37). These results in the framework of a Morita context ring yield the following:

Lemma 3.19. If M = (A, M, N, B, φ, ψ) is a Morita context such that T (M) is NI (resp., weakly 2-primal or 2-primal) then A and B are also NI (resp., weakly 2-primal or 2-primal). 77

3.3.1 NI Morita Context Rings   N(A) 0   Let M = (A, M, N, B, φ, ψ) be a Morita context. Clearly,  , 0 N(B)     0 M 0 0      , and   are subsets of N(T (M)). Thus, if N(T (M)) is closed 0 0 N 0 under addition, which is the case when N(T (M)) is an ideal of T (M), then   N(A) M     ⊆ N(T (M)). (3.3.1) NN(B)

Theorem 3.20. Let M = (A, M, N, B, φ, ψ) be a Morita context. Then T (M) is NI if and only if A and B are NI rings and   N(A) M   N(T (M)) =   . (3.3.2) NN(B)

Proof. (⇒) By Lemma 3.19, A and B are NI rings. We now prove (3.3.2). By   N(A) M   hypothesis N(T (M)) is an ideal, so by (3.3.1) we get   ⊆ N(T (M)). NN(B)   a m a   a To show the reverse inclusion, let =   ∈ N(T (M)). Then aE1 = E1 E1 n b and bE2 = E2aE2 are nilpotent elements of T (M) because N(T (M)) is an ideal of   N(A) M a   T (M). Hence a ∈ N(A) and b ∈ N(B). This shows that ∈  . NN(B) (⇐) If A and B are NI rings and (3.3.2) holds, then N(T (M)) is an ideal of T (M), whence T (M) is an NI ring.

Corollary 3.21. Let M = (A, M, N, B, φ, ψ) be a Morita context such that T (M) has zero trace ideals. Then T (M) is NI if and only if A and B are NI rings.

Proof. (⇒) If T (M) is NI then A and B are NI by Theorem 3.20. 78

(⇐) Assume that A and B are NI rings. In light of Theorem 3.20, it is enough to     N(A) M a m   a   show that N(T (M)) =  . Let =   ∈ T (M). Since T (M) NN(B) n b has zero trace ideals, it is straightforward to verify that for all integers k ≥ 1 we   k a mk k have a =   for some mk ∈ M and nk ∈ N. It follows that a ∈ N(T (M))  k  nk b   N(A) M a   if and only if ∈  . NN(B)

In [59], Hwang et al. proved that the ring Un(R) is NI if and only if R is NI,   AM   and that the triangular matrix ring   is NI if and only if A and B are NI 0 N rings. These results are now particular cases of Corollary 3.21.

Corollary 3.22. Let M = (A, M, N, B, φ, ψ) be a Morita context. If T (M) is NI, then A, B are NI rings, MN ⊆ N(A) and NM ⊆ N(B).

Proof. By Theorem 3.20, A and B are NI rings and (3.3.2) holds. Therefore    2   0 M 0 M MN 0         ⊆ N(T (M)), and so   =   ⊆ N(T (M)). N 0 N 0 0 NM

Example 3.23. From Corollary 3.22 we conclude that Mn(R) is non-NI for all rings R (which are assumed to have identity). Therefore, by Theorem 3.20, the   M (R) Rn×1  n  triangular matrix ring T =   is non-NI. Furthermore, T is 0 R non-reflexive by Example 3.10. The case n = 2 and F = F2 has been shown in [6, Theorem 2.7] to be the smallest non-reflexive non-NI ring.

It is worth noting that the converse of Corollary 3.22 is equivalent to the K¨othe’sconjecture, whose formulation is given below. To prove this fact we quote the following result due to A.D. Sands [86]. 79

Theorem 3.24 ([86, Theorem 10]). The following are equivalent: 1. K¨othe’sconjecture holds: for all rings R, Nil∗(R) contains every nil left or right ideal.

2. for some integer n ≥ 2, if R is a nil ring so also is Mn(R).

Before we prove that that the converse of Corollary 3.22 holds, we observe the following.

Remark 6. In the proof of Theorem 3.20 we used the fact that A and B are rings

with identity to conclude that aE1 = E1aE1 and bE3 = E2aE2 are nilpotent elements of T (M). The same conclusion can be drawn from the identities

3 3 a E1 = (aE1)a(aE1) and b E2 = (aE2)a(aE2). Hence Theorem 3.20 and its corollaries hold for arbitrary rings A and B (possibly without identity).

Proposition 3.25. The following are equivalent.

1. For some integer n ≥ 2, if R is a nil ring so also is Mn(R).

2. For all Morita context M = (A, M, N, B, φ, ψ), if A and B are NI rings (possibly without identity), MN ⊆ N(A) and NM ⊆ N(B) then T (M) is an NI ring.

Proof. 1) ⇒ 2) In view of Theorem 3.20 we only need to show that (3.3.2) holds.   N(A) M   ∗ First,   is an ideal of T because N(A) = Nil (A) and NN(B)   a m ∗   N(B) = Nil (B). Let X =   ∈ N(T (M)) and observe that n b         a 0 0 m 0 m N(A) M         X =   +  . Because   ∈   and 0 b n 0 n 0 NN(B) 80   N(A) M     is an ideal of T (M), then by induction on k ≥ 1 it is easy to NN(B)     ak 0 N(A) M k     prove that X =   + Yk, where Yk ∈  . Since X ∈ N(T ), 0 bk NN(B)   0 0 k   then X =   for some k, where it follows that a ∈ N(A) and b ∈ N(B). This 0 0   N(A) M   shows that N(T (M)) ⊆  . To prove the reverse inclusion, note that NN(B)       N(A) M N(A) 0 0 M         =   +  . It can easily be shown that NN(B) N 0 0 N(B)     N(A) 0 0 M       and   are left nil ideals of T (M). Therefore, because of 1) N 0 0 N(B) is equivalent to K¨othe’sConjecture by Theorem 3.24, we get   N(A) M   ∗   ⊆ Nil (T ) ⊆ N(T ). NN(B) 2) ⇒ 1) Let R be a nil ring. Then R is NI and from the hypothesis one can easily see that M2(R) is NI. Remark6 and Theorem 3.20 yield N(M2(R)) = M2(R), which

shows that M2(R) is nilpotent.

The proof of Proposition 3.25 implies the following useful corollary.

Corollary 3.26. Let M = (A, M, N, B, φ, ψ) be a Morita context such that A and B are NI rings (possibly without identity), MN ⊆ N(A) and NM ⊆ N(B). Then     N(A) M N(A) M       is an ideal of T (M) and N(T (M)) ⊆  . NN(B) NN(B) 81   F [x]/(xn) F   Example 3.27. Let T (M) =   be the Morita context ring FF [x]/(xn) of Example 3.11. Clearly, the ring A = F [x]/(xn) is NI and N(A) = (x). Moreover, since φ : F ⊗ F → A and ψ : F ⊗ F → A are given by φ : m ⊗ n 7→ mnx and   N(A) F   ψ : n ⊗ m 7→ nmx, then by Corollary 3.26, N(T (M)) ⊆  . The FN(A)   N(A) F   reverse inclusion is obvious, so N(T (M)) =  . By Theorem 3.20, FN(A) T (M) is NI, and so T (M) is a reflexive NI ring by Example 3.11. In [6, Theorem 3.7], it is shown that T (M) is a reflexive NI ring of minimal order when n = 2 and

F = F2.

3.3.2 Weakly 2-primal and 2-primal Cases

We now turn our attention to the study of weakly 2-primal and 2-primal Morita context rings. Recall that a ring R is called weakly 2-primal if

N(R) = L(R), while R is said to be 2-primal if N(R) = Nil∗(R). Radicals of a Morita context ring, like the Levitzki radical L(·) and the lower nilradical Nil∗(·), have been described in detail by A. D. Sands [86]. The statement for the radicals aforementioned is as follows.

Lemma 3.28 ([86, Theorem 8]). Let M = (A, M, N, B, φ, ψ) be a Morita context and N(·) one of the radicals L(·) or Nil∗(·). Then   N(A) M0 N   (T (M)) =   , N0 N(B) where M0 = {m ∈ M : Nm ⊆ N(B)} and N0 = {n ∈ N : Mn ⊆ N(A)}. 82

As an application of the previous result and Theorem 3.20 we obtain necessary and sufficient condition for a Morita context ring to be weakly 2-primal (resp., 2-primal).

Theorem 3.29. Let M = (A, M, N, B, φ, ψ) be a Morita context. The following conditions are equivalent 1. T (M) is weakly 2-primal (resp., 2-primal).

2. A, B are weakly 2-primal (resp., 2-primal), MN ⊆ N(A) and NM ⊆ N(B). Furthermore, if T (M) is weakly 2-primal (resp., 2-primal), then   N(A) M   N(T (M)) =   . NN(B)

Proof. We only prove the result for weakly 2-primal rings since the arguments carry over mutatis mutandis for the 2-primal case. Let T = T (M). 1) ⇒ 2) It follows from Lemma 3.19 and Corollary 3.22 since weakly 2-primal rings are NI.     L(A) M N(A) M  0    2) ⇒ 1) By Lemma 3.28, L(T ) =   =   ⊆ N(T ). N0 L(B) NN(B)   a m a   To see that the reverse inclusion is also true, let =   ∈ N(T ). Then n b       a 0 0 m 0 m a       =   +  , and observe that   ∈ L(T ). Since L(T ) is an ideal 0 b n 0 n 0   ak 0 ak   of T , an inductive argument on k ≥ 1 yields =   + Yk for some 0 bk k Yk ∈ L(T ). We have chosen a in N(T ), so a = 0 for some k ≥ 1. For this value of k   ak 0   we obtain   ∈ L(T ), and so a ∈ N(A) and b ∈ N(B). Consequently 0 bk N(T ) ⊆ L(T ). Therefore T is weakly 2-primal. 83

The following result is an immediate consequence of Theorem 3.29.

Corollary 3.30. Let M = (A, M, N, B, φ, ψ) be a Morita context. If T (M) has zero trace ideals, then T (M) is weakly 2-primal (resp., 2-primal) if and only if A and B are weakly 2-primal (resp., 2-primal).   AM   By Corollary 3.30, the triangular matrix ring   is weakly 2-primal 0 N (resp., 2-primal) if and only if A and B are weakly 2-primal (resp., 2-primal).

Consequently, for a ring R, Un(R) is weakly 2-primal (resp., 2-primal) if and only if R is weakly 2-primal (resp., 2-primal). In [9], Birkenmeier et al. derived these

results for the family of 2-primal rings. In [64], Chen and Cui proved that Un(R) is weakly 2-primal if and only if R is weakly 2-primal. Thus Corollary 3.30 can be considered as a generalization of these results. Finally, observe that the Morita context ring in Example 3.27 is weakly 2-primal and 2-primal by Theorem 3.29. 84 4 McCoy Rings and Polynomial Semicommutative Rings

4.1 An Overview on McCoy Rings

N. H. McCoy [89] proved in 1942 the landmark result that if f(x) ∈ R[x] is a zero divisor and R is commutative ring then there exists a nonzero r ∈ R such that f(x)r = 0. E. Armendariz [90] showed in a lemma that for a reduced ring R and

P i P j two polynomials f(x) = aix ∈ R[x], g(x) = bjx ∈ R[x], f(x)g(x) = 0 if and only if aibj = 0 for all i, j. Using this result, M. B. Rege and S. Chhawchharia [4]

P i P j called a ring R Armendariz if whenever f(x) = aix , g(x) = bjx ∈ R[x] satisfy f(x)g(x) = 0 then aibj = 0 for all i, j. In the same paper, McCoy rings are defined as follows.

Definition 4.1. A ring R is called right McCoy if whenever f(x) is a left zero divisor in R[x] there exists a non-zero element r ∈ R such that f(x)r = 0. Left McCoy rings are defined dually. A ring is a McCoy ring if it is both left and right McCoy.

The relation between McCoy rings and the other standard classes of rings defined in Chapter1 has been amply investigated (e.g. [12], [1], [13], [14], [15], [16], [3]). Rege and Chhawchharia [4] noted that Armendariz rings are McCoy. Nielsen [3] proved that reversible rings are McCoy. Camilo and Nielsen [1] showed that a right duo ring is also right McCoy. Remarkably, not all semicommutative rings are McCoy as it was shown by Nielsen [3]. However, Hirano [14, Theorem 2.2] proved that if R[x] is semicommutative then R is a McCoy ring. It was observed by Huh et al. [18], that there is a semicommutative ring R such that R[x] is not semicommutative. Thus the class of rings which have a semicommutative polynomial ring deserves special attention. We then introduced the notion of a polynomial semicommutative ring. 85

Definition 4.2. A ring R is said to be polynomial semicommutative if R[x] is semicommutative.

In this new terminology, polynomial semicommutative rings are McCoy rings by Hirano’s result [14, Theorem 2.2]. Some variations of the McCoy property have also been investigated. In [15], Hong et al. imposed an additional condition on the nonzero element r ∈ R in the McCoy property introducing the concept of strongly right McCoy rings.

Definition 4.3. A ring R is strongly right McCoy if for nonzero f(x), g(x) ∈ R[x], the condition f(x)g(x) = 0 implies f(x)r = 0 for some nonzero r in the right ideal of R generated by the coefficients of g(x).

Clearly, strongly right McCoy rings are right McCoy, but the converse need not to hold by [15, Example 9]. In addition, Hong et al. [15] proved that the classes of Armendariz, reversible and right duo rings are properly contained in the class of strongly right McCoy rings. Motivated by the results of Hong et al. [15] on strongly McCoy rings, Armendariz, reversible and right duo rings, Kim and Lee [16] introduced the notion of Armendarizlike rings. A ring R is said to be right Armendarizlike if for any nonzero f(x), g(x) ∈ R[x], f(x)g(x) = 0 implies there is r ∈ R such that g(x)r 6= 0 and f(x)gir = 0 for all coefficients gi of g(x). As one expects, right Armendarizlike rings are strongly right McCoy. Moreover, Armendariz, reversible and right duo rings are also right Armendarizlike [16]. In this chapter we expand the theory of McCoy rings and the related notions mentioned above in four directions. First, we investigate a special case in the definition of strongly right McCoy rings, which occurs when the nonzero element r is a right multiple of a single 86

coefficient of g(x). We call such rings right real McCoy rings. We show that a ring is right real McCoy if and only if it is right Armendarizlike. Due to this result, henceforth we refer to Armendarizlike rings as right real McCoy rings. Also, in relation to this result, we rise the question whether the concepts of strongly right McCoy and right real McCoy differ. We prove that [16, Example 1.9], which has been claimed to be an example of a strongly right McCoy but not right real McCoy, is flawed (see Example 4.7). Unfortunately, we have not found conclusive results, so we raise the question whether the classes of right real McCoy and strongly McCoy rings are different. Second, we show that the class of abelian right McCoy rings properly contains the classes of Armendariz, reversible, right duo, and polynomial semicommutative rings. Furthermore, we prove that neither the class of strongly right McCoy rings nor the class of semicommutative rings embraces all Armendariz, reversible, right duo, and polynomial semicommutative rings. Third, we provide a series of examples of finite local rings which are polynomial semicommutative. These examples, in addition of being polynomial semicommutative, have also any combination of the Armendarizness, right duoness and reversibility ring properties. The only combination we have not identified is a finite local ring which is Armendariz right duo but not reversible. We also determine whether the family of polynomial semicommutative rings is closed under polynomial ring formations. Although, semicommutative rings do not possess that property, we show that polynomial semicommutative rings do. Finally, we observe that the right McCoy, strongly right McCoy and right real McCoy ring properties, formulated in terms of polynomials in one variable, can be stated with respect to polynomials in two variables. 87

This chapter is organized as follows: Section 4.2 focuses on right real McCoy rings; Section 4.3 shows that the class of abelian right McCoy rings properly includes all Armendariz, reversible, right duo, and polynomial semicommutative rings; Section 4.4 discusses the various examples of finite local rings which are polynomial semicommutative rings; and Section 4.5 presents the new formulations of the right McCoy, strongly right McCoy and right real McCoy ring properties.

4.2 Right Real McCoy Rings

In this section we focus on a special case in the definition of strongly right McCoy rings, which results when the nonzero element r ∈ R is in a principal ideal of R rather than in a finitely generated ideal.

Definition 4.4. A ring R is said to be right real McCoy if for any nonzero f(x), g(x) ∈ R[x], f(x)g(x) = 0 implies f(x)r = 0 for some nonzero r in a principal right ideal of R generated by a coefficient of g(x). Left real McCoy rings are defined symmetrically. A ring that is both left and right real McCoy is called a real McCoy ring.

Clearly, Armendariz rings are principal McCoy. A less trivial result is [15, Theorem 1.6] which shows that reversible rings are right real McCoy. Likewise, by the proof of [15, Theorem 1.11], right duo rings are right real McCoy. We collect these results in the next theorem.

Theorem 4.5. Armendariz, reversible and right duo rings are right real McCoy.

Theorem 4.6. Let R be a real right McCoy ring. Then for any f(x), g(x) ∈ R[x],

f(x)g(x) = 0 implies there is an r ∈ R such that g(x)r 6= 0 and f(x)gir = 0 for

every coefficient gi of g(x). 88

Proof. Assume that R is a right real McCoy ring. Let f(x), g(x) ∈ R[x] be nonzero

Pn i such that f(x)g(x) = 0 and suppose that g(x) = i=0 gix . Then there is s0 ∈ {0, . . . , n} and a nonzero gs0 r0 ∈ gs0 R such that f(x)gs0 r0 = 0. If f(x)gir0 = 0 for any i, then r0 is the element for the described property. If not, then g(x)r0 6= 0

s0 and let g1(x) = g(x)r0 − gs0 r0x . Note that g1(x) 6= 0 and

0 = f(x)g(x)r0 = f(x)g1(x). Thus, there is s1 ∈ {0, . . . , n}\{s0} and a nonzero

(gs1 r0)r1 ∈ (gs1 r0)R such that f(x)gs1 r0r1 = 0. If f(x)gir0r1 = 0 for any i then r0r1 is the element element for the described property. If not, then g(x)r0r1 6= 0, define

s1 g2(x) = g1(x)r0r1 − gs1 r0r1x , and proceed as before to produce s2 ∈ {0, . . . , n}\{s0, s1}. Since the number of coefficients of g(x) is finite, this process will terminate in a finite number of steps, say k, producing r0r1 ··· rk 6= 0 such that g(x)r0r1 ··· rk 6= 0 and f(x)gir0 . . . rk = 0 for any i. Hence, R has the described property.

The property shown to be possessed by right real McCoy rings in Theorem 4.6 is precisely the Armendarizlike property defined above. Although the right real

McCoy condition is weaker as the difference is that the “for every coefficient gi of g(x)” in the definition of an Armendarizlike ring would be “for some coefficient gi of g(x)” in the definition of a right real McCoy ring, the two ring classes are the same by Theorem 4.6. Considering this remark, it is natural to ask whether the notions of strongly right McCoy and right real McCoy differ. The ring in [16, Example 1.9] has been claimed to be a right strongly McCoy ring that is not right real McCoy. However, the fact that the ring in [16, Example 1.9] is not right real McCoy is because the ring is not strongly McCoy as will be shown next. 89

Example 4.7 ([16, Example 1.9]). Let S = F2ha0, a1, a2, b0, b1, b2i and consider the S4 ring R = S/I, where I is the ideal of S generated by i=1 Ai with

A1 = {a0b0, a0b1 + a1b0, a0b2 + a1b1 + a2b0, a1b2 + a2b1, a2b2},

A2 = {xy | x ∈ {a0, a1, a2, b0, b1, b2}, y ∈ {a0, a1, a2}},

A3 = {xyz | x, y, z ∈ {b0, b1, b2}},A4 = {a1b0b2, a2b0b2}.

By abuse of notation, we identify s + I ∈ R with s ∈ S. From the relations in A1 and A2 we obtain that ai, bj are nilpotent in R. Thus, R is a finite local ring with

J = J(R) = (a0, a1, a2, b0, b1, b2). Consider the nonzero polynomials f(x), g(x) ∈ R[x]:

2 f(x) = a0 + a1x + a2x and g(x) = b1b2 + b2b2x.

From the relations in A1 we get that f(x)g(x) = 0. By the relations in A2 and A3, the right ideal generated by the coefficients of g(x) is

b1b2R + b2b2R = {0, b1b2, b2b2, b1b2 + b2b2}.

Since f(x)b1b2 6= 0, f(x)b2b2 6= 0 and f(x)(b1b2 + b2b2) 6= 0, it follows that R is not strongly right McCoy.

In [15, Section 2], Hong. et al. provide various simple ways to construct strongly right McCoy rings from any given strongly right McCoy ring. These constructions have been shown to yield right real McCoy rings from any given right real McCoy ring by Kim and Lee in [16]. However, none of these constructions can be used to construct a ring that differentiates the classes of strongly right McCoy and right real McCoy rings as we need to start with a strongly right McCoy ring which is not a right real McCoy ring. Although it seems that the right real McCoy property is stronger than the right strongly McCoy property, as of yet, we do not 90 know whether these two properties are equivalent. Considering this remark, we pose the next question.

Question 4.2.1. Does there exist a strongly right McCoy ring which is not right real McCoy?

In Example 1.10 in [15], Hong et al. also provide an example of a strongly McCoy ring which is not abelian. It was shown that this ring is strongly right McCoy because it is right real McCoy in [16, Example 2.2]. We conclude this section providing another example of a nonabelian right real McCoy ring.

K[t] Example 4.8. Let K be a field, S = (t2) = {s0 + s1t | s0, s1 ∈ K}, and consider K as a (K,S)-bimodule, where K K is the regular left module and k(s0 + s1t) = ks0 for all k ∈ K and s0 + s1t ∈ S. Let R be the formal upper triangular matrix ring   KK   R =   . 0 S

  1 0   Since e =   is a noncentral idempotent, R is nonabelian. We now show that R 0 0 Pn i Pm j is right real McCoy. Let F (x) = i=0 Aix , G(x) = j=0 Bjx be nonzero in R[x],     (i) (i) (j) (j) a11 a12 b11 b12 where Ai =   and Bj =  . Observe that F (x) and  (i) (i)  (j) (j) 0 a21 + ta22 0 b21 + tb22 G(x) can be written as     f (x) f (x) g (x) g (x)  11 12   11 12  F (x) =   and G(x) =   , 0 f21(x) + tf22(x) 0 g21(x) + tg22(x) 91

Pn (i) i Pm (j) j where fκ`(x) = i=0 aκ` x and gκ`(x) = j=0 bκ` x for 1 ≤ κ, ` ≤ 2. Assume that   0 0   F (x)G(x) = 0 and let et =  . Then 0 t

    f (x)g (x) f (x)g (x) + f (x)g (x) 0 0  11 11 11 12 12 21      =   . 0 f21(x)g21(x) + t[f21(x)g22(x) + f22(x)g21(x)] 0 0

This implies that f21(x) = 0 or g21(x) = 0.

Case 1: Assume g21(x) 6= 0.

(j) Then there is j, 1 ≤ j ≤ m, such that b21 6= 0. Thus, F (x)Bjet = 0 and observe that Bjet 6= 0 is in the right ideal generated by the coefficients of G(x).

Case 2: Assume g21(x) = 0.

Since F (x)G(x) = 0, this implies that f11(x) = 0 or g11(x) = 0.

Case 2(a): Assume f11(x) 6= 0.

From F (x)G(x) = 0 we have g11(x) = g12(x) = 0. In addition, since G(x) 6= 0 then f21(x) = 0. In this case, for any Bj 6= 0 we have F (x)Bj = 0.

Case 2(b): Assume f11(x) = 0.

This implies that f21(x) = 0 or g22(x) = 0. In either case, any Bj 6= 0 satisfies

F (x)Bj = 0.

The previous analysis shows that R is right real McCoy.

Remark 7. Recall that a ring is called abelian if every idempotent is central. Armendariz, reversible, and semicommutative rings are examples of abelian rings. Thus, Example 4.8 shows that the class of right real McCoy rings properly contains the classes of Armendariz, reversible, and right duo rings. 92

4.3 Abelian McCoy Rings

Hirano [14, Theorem 2.2] proved that if R is polynomial semicommutative then R is a McCoy ring. Thus each of the four ring theoretic properties, Armendarizness, reversibility, polynomial semicommutativity and right duoness, implies the right McCoy condition. A natural question is whether there is a sub-class of right McCoy rings which encompasses all these ring classes. In this section, we prove that the class of abelian right McCoy rings properly contains the classes of reversible rings, right duo rings, Armendariz rings and polynomial semicommutative rings. It has been shown by Chen et al. [13] that the class of semicommutative right McCoy rings properly contains both the classes of reversible rings and polynomial semicommutative rings. However, this class does not contain all Armendariz rings because there are Armendariz rings which are not semicommutative (see [18, Example 14]). On the other hand, in Theorem 4.5 we show that the class of right real McCoy rings contains the classes of Armendariz, reversible and right duo rings. Observe that this inclusion is proper by Remark7. It is natural to conjecture that polynomial semicommutative are right real McCoy. However, we have a negative answer to this situation in the following example. Moreover, the example shows that polynomial commutative rings are not necessarily strongly right McCoy rings.

Example 4.9. Let S = F2ha0, a1, b0, b1i and consider the ring R = S/I, where I is an ideal generated by the set of relations:

   xy | x, y ∈ {a0, a1, b0, b1} ∪ a0b1 + a1b0 \{a0b1, a1b0}.

We identify s + I with s ∈ S for all s + I ∈ R. The ring R can be easily verified to

3 be local with J(R) = (a0, a1, b0, b1) and J(R) = 0. By Proposition 4.12 below, R is polynomial semicommutative. We now prove that R is not strongly right McCoy. 93

To this aim, note that (a0 + a1x)(b0 + b1x) = 0. Also, note that b0R + b1R = {0, b0, b1, b0 + b1},(a0 + a1x)b0 = a1b0x 6= 0, (a0 + a1x)b1 = a0b1 6= 0, and (a0 + a1x)(b0 + b1) = a0b1(1 + x) 6= 0. Therefore, R is not right strongly McCoy.

In light of the previous example, a sub-class of right McCoy rings that contains Armendariz, reversible, right duo and polynomial semicommutative rings cannot be restricted to right real McCoy rings nor even strongly right McCoy rings. Semicommutative rings are ableian. Armendariz rings are also abelian by the method in the proof of [91, Theorem 6]. Although, it has been shown by Camilo and Nielsen that right McCoy rings are not necessarily abelian in [1, Theorem 7.1]. However, right McCoy rings are Dedekind-finite by [1, Theorem 5.2]. A diagram that summarizes the hierarchy of the implications between the ring classes discussed in the current and previous sections is given in Figure 4.1.

pol. semicomm.

? r. duo semicomm. abelian Dedekind-finite ?

reversible r. real McCoy str.r. McCoy r. McCoy

Armendariz Figure 4.1: McCoy rings

In the diagram in Figure 4.1, no other implications are possible, except for the ones indicated by the dashed arrows. The question whether right duo rings are polynomial semicommutative was originally posed in [12, Question 4.1], while whether strongly right McCoy rings are right real McCoy is Question 4.2.1. 94

The diagram shows that the class of abelian right McCoy rings contains the classes of reversible, Armendariz, right duo and polynomial semicommutative rings. As we pointed out before, either the class of strongly right McCoy rings nor the class of semicommutative rings embraces all Armendariz, reversible, right duo, and polynomial semicommutative rings. The next example is of a semicommutative right real McCoy ring (so abelian right McCoy) which lacks of the ring properties of reversibility, Armendarizness, duoness, and polynomial semicomutativity. This shows that the family of abelian right McCoy rings propertly contains the classes of reversible, Armendariz, right duo, and polynomial semicommutative rings.

Example 4.10. The ring in this example is from [18, Example 2]. Let

S = F2ha0, a1, a2, b0, b1, b2, ci and R = S/I, where I is the ideal of S generated by

the relations a0b0, a0b1 + a1b0, a0b2 + a1b1 + a2b0, a2b2, a0rb0, a2rb2,

(a0 + a1 + a2)r(b0 + b1 + b2), and r1r2r3r4, where

r, r1, r2, r3, r4 ∈ {a0, a1, a2, b0, b1, b2, c}. For simplicity, identify s + I ∈ R with s ∈ S.

Notice the ring R is a finite local ring with J(R) = (a0, a1, a2, b0, b1, b2, c) and J(R)4 = 0. In [18], Huh et al. proved that R is a semicommutative ring which is not polynomial semicommutative. By Lemma 4.13, R is not an Armendariz ring.

Observe that a0b0 = but b0a0 6= 0, so R is not reversible. Likewise, b0a0 ∈/ a0R, so R is not a duo ring. It was shown in [13, Example 5] that R is right McCoy but we will now show here that R possesses the stronger property of being right real McCoy.

Let f(x), g(x) ∈ R[x] be nonzero and assume f(x)g(x) = 0. Let g0 be some

t nonzero coefficient of g(x) and t be largest integer such that g0c 6= 0. Then

t 3 t g0c ∈ J(R) \ 0, so f(x)g0c = 0. Hence, R is right real McCoy. 95

4.4 Polynomial Semicommutative Rings

In this section a series of examples will be given to show that polynomial semicommutativity is independent of any combination of reversibility, Armendarizness, and duoness. All of the examples presented in this section are finite local polynomial semicommutative rings. They will be labeled with an “A” or “a”, a “R” or “r”, and a “D” or “d” depending on whether or not they are Armendariz, reversible and right duo, respectively. For example, “ARD” will be a polynomial semicommutative Armendariz reversible duo ring, while “ARd” will be a polynomial semicommutative Armendariz reversible ring which is not right duo. These are the first two examples, but before we need some basic facts about finite local rings and some preliminary results. Let R be a finite local ring, J = J(R) and F = R/J. Let k be the index of nilpotency of J, i.e., J k+1 = 0 and J k 6= 0. Then it is well-known that for each 1 ≤ i ≤ k, J i/J i+1 is a finite dimensional F -vector space. Thus, for each 1 ≤ i ≤ k,

i i+1 there exists a set of generators, say {ui1, . . . , uini } ⊂ J \ J , where

i i+1 i i+1 ni = dimF (J /J ). Thus every element of J /J can be written uniquely as

i+1 αi1ui1 + ··· + αini uini + J , where αij ∈ F . As a result, each element r ∈ R can be written uniquely as

r = α0 + α11u11 + ··· + α1n1 u1n1 + ··· + αk1uk1 + ··· + αknk uknk ,

where α0, αij ∈ F . Thus representation extends to the polynomial ring R[x], so every f(x) ∈ R[x] can be uniquely written as

f(x) = f0(x) + f11(x)u11 + ··· + f1n1 (x)u1n1
+ ··· + fk1(x)uk1 + ··· + fknk (x)uknk , 96

where f0(x), fij(x) ∈ F [x] for all 1 ≤ i ≤ k and 1 ≤ j ≤ ni. In other word, this shows that R is an F -algebra with basis {1} ∪ {uij | 1 ≤ i ≤ k, 1 ≤ j ≤ ni}. This fact will be used freely through this section. The following results will be needed throughout to show that a ring is polynomial semicommutative.

Lemma 4.11 ([14, Lemma 2.1]). Let f(x) and g(x) be two elements of R[x]. Then f(x)Rg(x) = 0 if and only if f(x)R[x]g(x) = 0.

In [19, Proposition 3.1] it was noted that if R is a finite local ring with R/J(R) a prime field and J(R)3 = 0 then R is semicommutative. We now prove a stronger result showing that the ring is in fact polynomial semicommutative.

Proposition 4.12. Let R be a finite local ring, J = J(R), and F = R/J(R). If F is a prime field and J 3 = 0 then R is polynomial semicommutative.

2 Proof. There exists a set {u1, . . . , un} ⊂ J \ J such that J = (u1, . . . , un) and

2 n = dimF (J/J ). Let f(x), g(x) ∈ R[x] and write these polynomials as

f(x) = f0(x) + fu1 (x)u1 + ··· + fuk (x)un + t1(x),

g(x) = g0(x) + gu1 (x)u1 + ··· + guk (x)un + t2(x),

2 where f∗(x), g∗(x) ∈ F [x], ∗ ∈ {0, u1, . . . , un}, and t1(x), t2(x) ∈ J [x]. Then f(x)g(x) can be written as

n ! X
f(x)g(x) = f0(x)g0(x) + f0(x)gui (x) + fui (x)g0(x) ui + t(x) i=0

2 for some t(x) ∈ J [x]. Assume f(x)g(x) = 0. Then f0(x)g0(x) = 0, so f0(x) = 0 or

g0(x) = 0. First, if f0(x) = 0 and g0(x) 6= 0, then fu1 (x) = ··· = fuk (x) = 0 meaning

2 2 f(x) ∈ J [x]. Similarly, if g0(x) = 0 and f0(x) 6= 0, then g(x) ∈ J [x]. In the case 97

that f0(x) = 0 and g0(x) = 0, then f(x), g(x) ∈ J[x]. In any case, for any r = r0 + r1 ∈ R, where r0 ∈ F and r1 ∈ J, we have

f(x)(r0 + r1)g(x) = r0f(x)g(x) + f(x)r1g(x) = 0.

This shows that f(x)Rg(x) = 0. By Lemma 4.11, f(x)R[x]g(x) = 0, showing R is polynomial semicommutative.

The hypothesis in Proposition 4.12 of F = R/J being a prime field is not superfluous as there exists a finite local nonsemicommutative ring R with J(R)3 = 0 and F = R/J(R) not a prime field. For instance, the ring

F4[v, σ][u] hu2 + v2, uvi has been shown in [19, Example 4.3] to be such a ring. Recall that a ring R is called locally finite if every finite subset of it generates a finite multiplicative semigroup. Clearly, any finite ring is locally finite.

Lemma 4.13. A locally finite Armendariz ring is polynomial semicommutative. In particular, finite Armendariz rings are polynomial semicommutative.

Proof. Let R be a locally finite Armendariz ring. By [18, Proposition 16], a locally finite Armendariz ring is semicommutative. From [4, Proposition 4.6], an Armendariz semicommutative ring is polynomial semicommutative.

Example 4.14 (ARD). Let σ be the Frobenius automorphism on F4 and [u; σ] R = F4 . (u2) Then R is a local ring with J(R) = (u) and J(R)2 = 0. From [6, Theorem 3.8] we

deduce that R is a reversible duo ring. We write f(x) = f0(x) + fu(x)u and g(x) = g0(x) + gu(x)u where f0(x), fu(x), g0(x), gu(x) ∈ F4[x]. Thus

0 = f(x)g(x) = f0(x)g0(x) + f0(x)gu(x) + fu(x)g0(x) u. 98

This implies f0(x)g0(x) = 0, showing f0(x) = 0 or g0(x) = 0. If f0(x) 6= 0 and g0(x) = 0 then f0(x)gu(x) = 0, from where it follows that g0(x) = gu(x) = 0, i.e., g(x) = 0. Likewise, if f0(x) = 0 and g0(x) 6= 0 then fu(x) = 0, and so f(x) = 0.

2 Thus, we may assume that f(x) = fu(x)u and g(x) = gu(x)u. Since J(R) , it follows in this case that every coefficient of g(x) is in the right annihilator of f(x). This shows that R is Armendariz. By Lemma 4.13, R is polynomial semicommutative.

Example 4.15 (ARd). Consider ring

hu, vi R = F2 . (u3, v2, u2 + uv + vu, uvu)

Note that R is a local ring with J(R) = (u, v), J(R)2 = (uv, vu), and J(R)3 = 0. By [6, Theorem 3.9], R is a reversible non-duo ring. We now prove that R is polynomial semicommutative. By Lemma 4.13, it is enough to show that R is Armendariz. Let f(x), g(x) ∈ R[x] and write these polynomials as

f(x) = f0(x) + fu(x)u + fv(x)v + fuv(x)uv + fvu(x)vu,

g(x) = g0(x) + gu(x)u + gv(x)v + guv(x)uv + gvu(x)vu, where f∗(x), g∗(x) ∈ F2[x] and ∗ ∈ {0, u, v, uv, vu}. Assume f(x)g(x) = 0. Then f0(x)g0(x) = 0, so f0(x) = 0 or g0(x) = 0. As in Example 4.14, it can be shown by expanding the product of f(x)g(x) that if f0(x) = 0 and g0(x) 6= 0 then f(x) = 0.

Likewise, it can be shown that if f0(x) 6= 0 and g0(x) = 0 then g(x) = 0. Thus we now assume that f0(x) = g0(x) = 0. Therefore

0 = f(x)g(x) = (fu(x)gu(x) + fu(x)gv(x))uv + (fu(x)gu(x) + fv(x)gu(x))vu.

Case 1: Suppose fu(x) = 0.

Then fv(x) = 0 or gu(x) = 0. In either case, it is easy to see that any coefficient of g(x) is in the right annihilator of f(x). 99

Case 2: Suppose fu(x) 6= 0.

Then gu(x) = gv(x) and fu(x) = fv(x), which implies that f(x) = fu(x)(u + v) + fuv(x)uv + fvu(x) and

2 g(x) = gu(x)(u + v) + guv(x)uv + fvu(x)vu. Since (u + v) = 0, it follows that every coefficient of g(x) is in the right annihilator of f(x).

From Cases 1 and 2, we conclude that R is an Armendariz ring.

The ring in the next example has been claimed to be a non-reversible polynomial semicommutative ring in [3]. Since the ring R is local with J(R)3 = 0, by Proposition 4.12, it is polynomial semicommutative. We now show the stronger fact that R is Armendariz, where it also follows that R is polynomial semicommutative by Lemma 4.13. In addition, it will be shown that R is non-duo.

2 2 Example 4.16 (aRD). Let R = Z4[u]/(u ) and f(x) = 2 + ux. Then f(x) = 0 but 2u 6= 0. Hence R is not Armendariz. However, since R is commutative, it is reversible and polynomial semicommutative.

Example 4.17 (arD). Let

hu, vi R = F2 . (u2 + uv, vu, v2 + uv)

From [6, Theorem 3.1] we deduce that R a non-reversible duo ring. Since (v + ux)(u + (u + v)x) = 0 but u2 6= 0, R is not Armendariz. Finally, observe that R

∼ 3 is a local finite ring with J(R) = (u, v), R/J(R) = F2, and J(R) = 0. Therefore, by Proposition 4.12, R is a polynomial semicommutative ring.

Example 4.18 (aRd). Let

hu, vi R = F2 . (u3, v3, u2 + v2 + vu, vu2 + uvu + vuv, u2vu) 100

This ring was shown to be a non-duo reversible ring in [5, Example 4.1]. Since (uv + (u2 + v2 + uv)x + v2x2)((u + v) + vx) = 0 but uv2 = uv · v 6= 0, R is not an Armendariz ring. We now show that R is polynomial semicommutative. First, observe that R is a finite local ring with J = J(R) = (u, v), J 2 = (u2, v2, uv), J 3 = (uv2, u2v), and J 4 = 0. Thus f(x), g(x) ∈ R[x] can be written as

f(x) = f0(x) + fu(x)u + fv(x)v + h(x) and g(x) = g0(x) + gu(x)u + gv(x)v + k(x)

2 2 2 2 2 where f∗(x), g∗(x) ∈ F2[x], h(x), k(x) ∈ J [x], and ∗ ∈ {u, v, u , v , uv, u v, uv }.

Assume that f(x) 6= 0, g(x) 6= 0, and f(x)g(x) = 0. Then f0(x) = 0 and g0(x) = 0. Thus

2
2 0 = f(x)g(x) = fu(x)gu(x) + fv(x)gu(x) u + fv(x) gv(x) + gu(x) v
+ fu(x)gv(x) uv + f(x)k(x) + h(x)g(x).

This implies that fu(x)gv(x) = 0, so fu(x) = 0 or gv(x) = 0.

Case 1: Assume that fu(x) = 0.

Then fv(x) = 0 or gu(x) = 0. In either case it can be shown that

2 2 f(x) = k(x) ∈ J [x] or g(x) ∈ J [x]. Thus, for all r = r0 + r1 ∈ R, with r0 ∈ F2 and r1 ∈ J, we have f(x)rg(x) = r0f(x)g(x) + f(x)r1g(x) = 0.

Case 2: Assume that gv(x) = 0.

Then fv(x) = 0 or gu(x) = 0. As before, in either case it can be proved that f(x) = k(x) ∈ J 2[x] or g(x) ∈ J 2[x]. Thus f(x)rg(x) = 0 for all R ∈ R. In any case, by Lemma 4.11, since f(x)Rg(x) = 0, f(x)R[x]g(x) = 0. So, R is polynomial semicommutative.

Example 4.19 (ard). Let hu, vi R = Z4 . (u2, vu, v2, 2v) The ring R is non-reversible since vu = 0 but uv 6= 0. Also, the ring R is non-duo since uv 6∈ vR = {0, v}. Observe that R is a local ring with J = J(R) = (2, u, v), 101

2 3 ∼ J = (uv, 2u), and J = 0. Since R/J = F2 is a prime field, R is a polynomial semicommutative ring in light of Proposition 4.12.

The ring in Example 4.9 is a polynomial semicommutative ring which is not a right real McCoy ring. Hence, the ring in Example 4.9 cannot be a reversible, right duo and Armendariz ring (see Theorem 4.5). On the other hand, the ring in Example 4.19 is also a polynomial semicommutative ring which is not reversible, right duo nor Armendariz. Unlike Example 4.9, it can be shown that Example 4.19 is right real McCoy. We have produced 7 examples of polynomial semicommutative rings which in addition are or not are Armendariz, reversible and right duo. We do not know whether a polynomial semicommutative ring which is Armendariz and right duo but not reversible exists, so we pose the following question.

Question 4.4.1. Does there exist a finite ring which is Armendariz and right duo but not reversible?

Note that a finite Armendariz right duo ring would be polynomial semicommutative in view of Lemma 4.13. As we mentioned before, the class of semicommutative rings is not closed under polynomial ring formation. The next theorem states that the family of polynomial semicommutative rings does. We base the proof of this result on the proof of [91, Theorem 1], where it was shown that a ring R is Armendariz if and only if R[x] is Armendariz.

Theorem 4.20. A ring R is polynomial semicommutative if and only R[x] is polynomial semicommutative.

Proof. Assume R is polynomial semicommutative. Let F (y),G(y) be nonzero polynomials in R[x][y] with F (y)G(y) = 0. In view of Lemma 4.11, we only need to 102

n show that F (y)rG(y) = 0 for all r ∈ R[x]. Write F (y) = f0 + f1y + ··· + fny and

m G(y) = g0 + g1y + ··· + gmy where fi, gj ∈ R[x]. Let r ∈ R[x] and define Pn Pm k = 1 + i=0 deg fi + j=0 deg gj + deg r, where the degree is as polynomials in R[x] and the degree of the zero polynomial is taken to be zero. Then the set of

k coefficients of the fi’s (resp., gj’s) equals the set of coefficients of F (x ) (resp., G(xk)) in R[x]. Since x commutes with the elements of R and F (y)G(y) = 0, the selection of k implies that F (xk)G(xk) = 0. Since R is polynomial semicommutative then F (xk)rG(xk) = 0. The election of k implies that F (y)rG(y) = 0. Therefore R[x][y] is semicommutative. The converse follows from the fact that subrings of polynomial semicommutative rings are also polynomial semicommutative.

Corollary 4.21. A ring R is polynomial semicommutative if and only if

R[x1, . . . , xn] is polynomial semicommutative for all n ≥ 1.

Corollary 4.22. Let R be a ring and X = {xα} any collection of commuting indeterminates over R. Then R is polynomial semicommutative if and only if R[X] is polynomial semicommutative.

Proof. Let F (y),G(y) ∈ R[X][y] such that F (y)G(y) = 0. Let r ∈ R[X]. Then

F (y),G(y), r ∈ R[xα1 , . . . , xαn ][y] for some finite subset {xα1 , . . . , xαn } ⊆ X. The ring R[xα1 , . . . , xαn ] is polynomial semicommutative by Corollary 4.21. Hence F (y)rG(y) = 0, where it follows that R[X][y] is semicommutative. The converse is clear.

4.5 McCoy Conditions and Polynomial Rings

The connection between the different McCoy conditions on a ring R and the polynomial ring R[x] has been investigated by some authors [13, 15, 16]. In this section, we formulate equivalent conditions for the notions of right McCoy, right real 103

McCoy and strongly right McCoy in terms of zero divisors for polynomials in the ring R[x, y]. Due to Lei et al. [13], a ring R is McCoy if and only if R[x] is McCoy ([13, Theorem 1]). In fact, an analysis of the proof of this result yields that a ring R is right McCoy if and only if R[x] is right McCoy. Recall that R[x] is right McCoy if for every nonzero polynomials F (y),G(y) ∈ R[x][y] such that F (y)G(y) = 0 there is a nonzero r(x) ∈ R[x] such that F (y)r(x) = 0. Although R[x][y] ∼= R[x, y] and F (y),G(y) can be viewed as polynomials in R[x, y], note that the condition on R[x] of being right McCoy implies the existence of a nonzero polynomial r(x) ∈ R[x] in the right annihilator of F (y) rather than a nonzero element r ∈ R. However, we next show that it is always possible to choose r ∈ R.

Theorem 4.23. Let R be a ring. The following statements are equivalent.

1. R is right McCoy.

2. For all nonzero F (x, y),G(x, y) ∈ R[x, y], F (x, y)G(x, y) = 0 implies F (x, y)r = 0 for a nonzero r ∈ R.

3. R[x] is right McCoy.

Proof. By the preceding remarks, we only need to show that (1) ⇔ (2). (1) ⇒ (2) Let R be a right McCoy ring and let F (x, y),G(x, y) ∈ R[x, y] be nonzero

Pn i polynomials such that F (x, y)G(x, y) = 0. Write F (x, y) = i=0 fi(x)y and Pm j G(x, y) = j=0 gj(x)y , where fi(x), gj(x) ∈ R[x] for all 0 ≤ i ≤ n and 0 ≤ j ≤ m. Pn Pm k Let k = 1 + i=0 deg fi(x) + j=0 deg gj(x). Then substituting y by x yields

k k nk F (x, x ) = f0(x) + f1(x)x + ··· + fn(x)x ∈ R[x];

k k mk G(x, x ) = g0(x) + g1(x)x + ··· + gm(x)x ∈ R[x]. 104

Because k > deg fi(x) for all i, 0 ≤ i ≤ n, there are no like terms between any of the

ik k k fi(x)x in the expression of F (x, x ). Thus the substitution of y by x , and vice-versa, creates a one-to-one correspondence between F (x, y) and F (x, xk).

k jk Therefore F (x, y) 6= 0 if and only if F (x, x ) 6= 0. The same is true for the gj(x)x ’s

k and G(x, x ) because k > deg gj for all 0 ≤ j ≤ m. Since F (x, y)G(x, y) = 0 and x commutes with the elements of R, then F (x, xk)G(x, xk) = 0. Because R is right McCoy, F (x, xk) 6= 0, and G(x, xk) 6= 0, there exists a nonzero r ∈ R such that

k k k nk F (x, x )r = 0. That is, 0 = F (x, x )r = f0(x)r + f1(x)rx + ··· + fn(x)rx . It follows that F (x, y)r = 0. (2) ⇐ (1) This follows from the fact that R[x] ⊂ R[x, y].

Corollary 4.24. Let R be a ring and X = {xα} any non-empty collection of commuting indeterminates over R. Then R is right McCoy if and only if R[X] is right McCoy.

Proof. Assume that R is a right McCoy ring. Let F (t),G(t) be polynomials in

R[X][t] such that F (t)G(t) = 0. Then F (t),G(t) ∈ R[xα1 , . . . , xαn ][t] for some finite subset {xα1 , . . . , xαn } ⊆ X. A direct application of part 2 of Theorem 4.23 yields that R[xα1 ] is right McCoy. Thus an inductive argument shows that R[xα1 , . . . , xαn ] is right McCoy. Thus there exists a nonzero r ∈ R[xα1 , . . . , xαn ] ⊂ R[X] such that F (t)r = 0. Hence R[X] is right McCoy.

Pn i Assume that R[X] is a right McCoy ring. Let f(x) = i=0 fix and g(x) be nonzero polynomials in R[x] such that f(x)g(x) = 0. The substitution of the inderterminate x by t yields f(t), g(t) ∈ R[X][t] and f(t)g(t) = 0. Since R[X] is right McCoy, there is a nonzero r ∈ R[X] such that f(t)r = 0. This implies that fir = 0 for all 0 ≤ i ≤ n. Thus any nonzero coefficient of r, say ρ ∈ R, is in the right annihilator of fi for all 0 ≤ i ≤ n. Thus f(x)ρ = 0, which implies that R is right McCoy. 105

The parallel version of Theorem 4.23 for right real McCoy rings is given next. Furthermore, we give additional conditions motivated by the fact that the ring property defining Armendarizlike rings is equivalent to the property in the definition of right real McCoy rings.

Theorem 4.25. The following statements are equivalent for a ring R.

1. R is right real McCoy.

2. For all nonzero F (x, y),G(x, y) ∈ R[x, y] such that F (x, y)G(x, y) = 0 there is a nonzero element r in the right ideal of R generated by one coefficient of G(x, y).

P i j P k l 3. For all nonzero F (x, y) = i,j fijx y ,G(x, y) = k,l gk,lx y ∈ R[x, y] such that F (x, y)G(x, y) = 0 there is a nonzero r ∈ R such that G(x, y)r 6= 0 and

fi,jgk,lr = 0.

4. R[x] is right real McCoy.

Proof. The proof of (1) ⇔ (2) is similar to the proof of Theorem 4.23. We now show that (2) ⇒ (3). Assume (2). Then R is right real McCoy. In light of Theorem 4.6, R is right Armendarizlike. As in the proof of Theorem 4.23, it can be shown that R being right Armendarizlike implies (3). Clearly, (3) ⇒ (2). Finally, Kim and Lee proved in [16, Theorem 3.1] that a ring R is right Armendarizlike if and only if so is R[x]. This establishes that (1) ⇔ (4).

Corollary 4.26. Let R be a ring and X = {xα} any non-empty collection of commuting indeterminates over R. Then R is right real McCoy if and only if R[X] is right real McCoy.

Proof. Similar to Corollary 4.24. 106

We now present a characterization of strongly McCoy rings. Its proof is similar to the proof of Theorem 4.23.

Theorem 4.27. Let R be a ring. The following statements are equivalent.

1. R is strongly right McCoy.

2. For all nonzero F (x, y),G(x, y) ∈ R[x, y], F (x, y)G(x, y) = 0 implies F (x, y)r = 0 for a nonzero r in the right ideal of R generated by the coefficients of G(x, y).

Hong et al. [15] proved that if R[x] is strongly right McCoy then R is strongly right McCoy. We can recover this result from Theorem 4.27.

Corollary 4.28. If R[x] is strongly right McCoy then R so is.

The converse of the previous Corollary 4.28 was posed as an open question in [15]. UNfortunately, we have not found a conclusive answer either. Encompassing this, we formulate the following:

Question 4.5.1. For a ring R, Does Theorem 4.27(2) implies that R[x] is strongly right McCoy?

The diagram in Figure 4.2 summarizes Theorem 4.23, Theorem 4.25, and Theorem 4.27. According to the diagram in Figure 4.2, a negative answer to Question 4.2.1 would give a positive answer to Question 4.5.1. 107

R is r. Armendarizlike R is r. real McCoy R is str. r. McCoy R is r. McCoy

Theorem 4.25(3) Theorem 4.25(2) Theorem 4.27(2) Theorem 4.23(2)

R[x] r. Armendarizlike R[x] r. real McCoy R[x] str. r. McCoy R[x] r. McCoy Figure 4.2: The McCoy conditions and polynomial rings 108 5 Reflexive non-Dedekind-finite Rings

According to the diagram in Figure 1.1, various of the classes of interest here can be encompassed in two big categories: reflexive rings and Dedekind finite rings. It is easy to see that finite rings are Dedekind-finite. Thus many of the examples of rings presented in this dissertation are Dedekind-finite. So, any finite non-reflexive ring serves as an example of a Dedekind-finite non-reflexive ring. For example, the ring F2hu, vi (v2, vu, u2 − uv) is a finite non-reflexive ring, so it is an example of a Dedekind-finite non-reflexive ring. However, as Noetherian rings are Dedekind-finite by [56, Exercise 1.12], no finite ring serves as an example of a non-Dedekind-finite ring. In addition, non-Dedekind-finite rings are necessarily non-Noetherian by the same exercise. Many examples of rings are known to be Noetherian, but there are also plenty of non-Noetherian rings. The classical example of a non-Dedekind-finite ring is the following (see [56, pp. 4 ]).

Example 5.1. Let F be a field and V be the F -vector space with a countable infinite basis B = {ei : i ≥ i}. Let R = EndF (V ) be the F -algebra of all vector

space endomorphisms of V . Let 1V denote the identity endomorphism on V, i.e., the identity of the ring R. Let ψ, φ ∈ R be the defined on the basis B of V as follows:

ψ(e1) = 0, φ(ei) = ei−1for all i ≥ 2, and

φ(ei) = ei+1 for all i ≥ 1.

Then φ ◦ ψ = 1V but ψ ◦ φ 6= 1V . Hence R is a non-Dedekind-finite ring.

In this chapter, we generalize the previous example and show that the ring CFM(R) of all column finite matrices and the ring RCFM(R) of all row and 109

column finite matrices over a ring R is reflexive if and only if R is reflexive. Since RCFM(R) and CFM(R) are non-Dedekind-finite rings, this result lands a handfull of examples of non-Dedekind-finite rings.

5.1 Row and Column Finite Matrices

Let V and W vector spaces over a field F , and let β and β0 be bases for V and W , respectively. In Linear Algebra we learn that if V and W are finite dimensional then any linear transformation τ : V → W can be represented by a matrix of size m × n that depends on the bases β and β0, where m = |β0| and n = |β|. Furthermore, we prove that the abelian group

L(V,W ) = {τ : V → W : τ is linear transformation}

is isomorphic to the abelian group Mm,n(F ) of all m × n matrices over F . In ∼ particular, when W = V , we prove that EndF (V ) = L(V,V ) = Mn(F ). If one of the vector spaces V and W is infinite dimensional, then we can still represent a linear transformation τ : V → W by a matrix, but in this case the matrix has infinite rows and/or columns. For example, let V = W = F [x], and let β = {1, 1 + x, 1 + x + x2, 1 + x + x2 + x3,...} and β0 = {1, x, x2, x3,...}. The matrix

0 associated to the identity map 1V : V → V with respect to the basis β and β is   1 1 1 1 ···     0 1 1 1 ···   β0   [1V ] = 0 0 1 1 ··· . β       0 0 0 1 ··· . . . . .  ......

β0 Observe that each entry of the first row of [1V ]β is equal to 1, while each column of β0 [IV ]β has a finite number of nonzero entries. A matrix whose columns have a only a 110

finite number of non-zero entries is called a column finite matrix. For example,

β0 [1V ]β is a column finite matrx. It is not difficult to prove that to each column finite matrix corresponds uniquely to a linear transformation between V and W , so we have a bijection

between L(V,W ) and the set CFMI,J (F ) of all column finite matrices of size I × J, where I = dim(W ) and J = dim(V ). Some algebraic operations as coordinate-wise addition, scalar multiplication, and transposition can still be defined without problem for infinite matrices; but matrix multiplication may involve infinite summations to define the resulting entries, and these are not defined in general. However, multiplication is well defined between two column finite matrices. Thus, with coordinate-wise addition and usual product of matrices, the set CFMI (F ) = CFMI,I (F ) is a non-commutative ring with identity (see [46, Proposition 1.14]). In fact, it can be proved that coordinate-wise addition of column finite matrices corresponds to addition of linear transformations, as well as product of such matrices to composition of linear ∼ transformations. That is, as rings, L(V,V ) = CFMI (F ). The above results generalize to free modules over associative rings. If R is an associative ring with identity, then any right R-homomorphism between two free right R-modules is pictured by a column finite matrix with entries in R. In fact, we

(I) prove that EndR(R ) and CFMI (R) are isomorphic rings (see [46, Exercise 21 of

§6]). Furthermore, if r ∈ R acts on A = (aij) ∈ CFM(R) as rA = (raij), then CFM(R) becomes a left R-module.

Let RCFM(R) be the subring of CFM(R) = CFMN(R) consisting of all matrices which are both row and column finite. Clearly, RCFM(R) is also a left

(N) ∼ submodule of CFM(R). A map f ∈ EndR(R ) = CFM(R) is said to be a continuous endomorphism if its matrix representation, which we denote by [f], is in 111

RCFM(R) for some basis B of M (see [92]). This is why the ring RCFM(R) of row and column finite matrices is also know as the ring of continuous endomorphisms of free modules. The ring RCFM(R) has also recently emerged in the study of countable dimensional algebras, rings without finiteness conditions, in the context of Morita equivalence, exchange rings and Baer rings (see [93] and the references included there). Recently, topological notions have been replaced by arithmetic tools and we often say “endomorphism with-adjoint” instead of continuous endomorphism to refer to those endomorphisms that have a matrix representation in RCFM(R) for a given basis B.

5.2 The Universal Property of Monoid Rings

Let R be a ring with identity 1 and (G, ∗) be a monoid with identity . Let R[G] be the free additive abelian group R(G) on the set G. That is, R[G] is the set

of all families a = {ag}g∈G such that ag ∈ R for all g ∈ G and ag = 0 for all but a finite number of indexes g ∈ G; equipped with component-wise addition, i. e., for

a = {ag}g∈G and b = {bg}g∈G in T [G] we have

a + b = {ag + bg}g∈G.

Clearly, the zero of T [G] is 0 = {0}g∈G, and the additive inverse of a = {ag}g∈G is

−a = {−ag}g∈G. Since R is a left R-module, the free abelian group R[G] can be turned into a left R-module by defining the action of r ∈ R on a = {ag}g∈G as ra = {rag}g∈G. Clearly, 1a = a for all a ∈ R[G], so R[G] is a unitary left R-module. Note that R[G] is also a unitary right R-module if we define ar = {agr}g∈G.

For each g ∈ G, let eg be the element {xλ}λ∈G ∈ R[G] such that xλ = 0 if λ 6= g and xλ = 1 if λ = g. Then, the set {eg : g ∈ G} is a basis of R[G], so every element 112

a ∈ R[G] admits a unique expression as a sum

X a = ageg. g∈G

Although this sum is taken over all elements of G, note that it is always finite as

ag = 0 for all but a finite number of indexes g ∈ G. By using such expressions for elements of R[G], we turn R[G] into a ring with P P identity by defining multiplication of a = g∈G ageg and b = h∈G bheh as

X ab = agbheg∗h. g,h∈G

The identity of the ring R[G] is 1e, which can be trivially verified. The ring R[G] is called the monoid ring or monoid algebra of G over R. Observe that R[G] is a commutative ring if and only both the ring R and the monoid (G, ∗) are commutative. We now embed R and G in a natural way in R[G].

Let α : R → R[G] be the map given by α(r) = r(1e) = re. It is immediately verified that α is a monomorphism from the ring R into R[G], so by the First Isomorphism Theorem, ∼ R = Im(α) = {re : r ∈ R}.

Thus, R[G] contains an isomorphic copy of R, whence we view R as a subring of R.

Let β : G → R[G] be the function given by β(g) = eg. Then

β(g ∗ h) = eg∗h = egeh = β(g)β(h), so β is a multiplicative monoid homomorphism, and is in fact injective, i.e., an embedding. Thus, R[G] contains an isomorphic copy of the monoid G, namely

∼ G = Im(β) = {eg : g ∈ G}. 113

Therefore, we can also say that G is a submoinoid of the monoid (R[G], ·). On the other hand, since

(re)(eg) = (re)(1eg) = (r1)e∗g = reg,

and similarly, (eg)(re) = (1eg)(re) = reg, we have that every element of Im(α) commutes with every element of Im(β), i. e. every element of R ⊂ R[G] commutes with every element of G ⊂ R[G]. The maps α and β characterize the monoid ring R[G] in the following sense.

Lemma 5.2 (Universal property of the monoid ring, [94, Proposition 3.2.7]). Let R be any ring with identity and (G, ∗) be a monoid. If S is a ring with identity, ϕ : R → S a ring homomorphism, and ψ :(G, ∗) → (S, ·) a monoid homomorphism such that ϕ(R) commutes with ψ(G), then there exists a unique ring homomorphism Φ: R[G] → S such that the diagrams commute

R[G] R[G] α Φ β Φ

ϕ ψ R S G S

5.3 A Monoid Ring Embedded in RCFM(R)

Let R be a ring M = R(N) the free right R-module. A basis of M is the set

B = {ei : i ∈ N}, where ei = (0,..., 0, 1, 0,...) and 1 is at the ith position. With P respect to this basis, any element m ∈ M can be uniquely written as m = i∈A eiri for some nonempty set A ⊂ N, and some ri ∈ R. Note that M is also an R-bimodule if we define

r(eiri) = (0,..., 0, rri, 0 ...) ∀ r ∈ R,

so that X rm = r(eiri). i∈A 114

Because R is not be commutative, we may have rm 6= mr for some m ∈ M and

r ∈ R. However, since r1 = 1r = r for all r ∈ R, then rei = eir for all i ∈ N. Thus, P P any element m ∈ M can be written uniquely as m = i∈A eiri or m = i∈A riei for

some nonempty set A ⊂ N, and some ri ∈ R.

Let EndR(M) be the set of all right R-module endomorphisms f : M → M.

We know that EndR(M) is a noncommutative ring with identity with respect of point-wise addition of maps and product given by composition (the identity being the identity map on M, which we denote by 1M ). The ring EndR(M) is usually called the endomorphism ring of M.

We now consider a subset of EndR(M). Remember that M is a free (right and left) R-module, so in order to define an endomorphism f : M → M it is enough to specify the action of f on the basis {ei : i ∈ N} of M. Let X : M → M and Y : M → M be the unique right endomorphisms of M such that

X(e1) = 0,X(ei) = ei−1 for all i ≥ 2, and

Y (ei) = ei+1 for all i ≥ 1.

It follows from the definitions of X and Y that XY = 1M but YX 6= 1M . Thus, the ring EndR(M) is a non-Dedekind finite ring.

Since tei = eit for all r ∈ R and i ∈ N, then

X(rei) = X(eir) = X(ei)r = rX(ei), and

Y (rei) = Y (eir) = Y (ei)r = rY (ei).

This implies that X and Y are also left R-endomorphisms of M. So X and Y are R-bimodule endomorphisms of M. However, since R is not necessarily commutative, we may have the case X(m)r 6= rX(m) for some m ∈ M and r ∈ R. 115

k Qk k Qk For each inetger k ≥ 1, let X = l=1 X and Y = l=1 Y . Because X and Y are R-bimodule endomorphisms of M, it follows that for all k ≥ 1, the maps Xk and Y k are also R-bimodule endomorphisms such that  0 if i ≤ k k  X (ei) = ,  ei−k if i ≥ k

k k k and Y (ei) = ei+k for all i ≥ 1. Thus, the kernel and the image of X and Y are

k k ker(X ) = he1, . . . , eki ⊂ M, Im(X ) = M,

and

k k ker(Y ) = {0}, Im(Y ) = he1+k, e2+k, e3+k,...i.

k k Therefore, for all k ≥ 1, we have X Y = 1M and  0 if i ≤ k k k  (Y X )(ei) = .  ei if i > k

k k k k Thus (1M − Y X )(ei) = ei if i ≤ k, and (1M − Y X )(ei) = 0 if i > k. Finally, for

different powers m, n ∈ N, we have   m−n X if m > n XmY n =  n−m Y if n < m and  0 if i ≤ n m n  (Y X )(ei) =  ei−n+m if i > n All these facts imply that the product of Y mXn and Y kXl yields another

a b element Y X ∈ EndR(M). This proves the following lemma, where by convention

0 0 we set X = 1M and Y = 1M . 116

m n Lemma 5.3. Let H = {Y X : m, n ≥ 0} ⊆ EndR(M). Then H is a monoid with respect to composition of endomorphisms.

With respect to the basis B = {ei : i ∈ N} of M, the matrix representation of X and Y are     0 1 0 0 ··· 0 0 0 0 ···         0 0 1 0 ··· 1 0 0 0 ···         [X] = 0 0 0 1 ··· and [Y ] = 0 1 0 0 ··· .             0 0 0 0 ··· 0 0 1 0 ··· . . . . .  . . . . .  ......

Since [X], [Y ] ∈ RCFM(R), then all elements in the monoid H in Lemma 5.3 have matrix representation in RCFM(R). In other words, all maps in the monoid H are continuous endomorphisms. The matrix representation of some elements in H is given next.

kthcol

↓       0 ··· 0 1 0 0 ··· [X]k = = =  0N,k−1 [X] 0N,k I   0 ··· 0 0 1 0 ···      0 ··· 0 0 0 1 ···    ...... 117

  0 0 0 0 ···   . . . .  . . . .          th 0k−1,N 0k,N 0 0 0 0 · · · ← k row k       [Y ] =   =   =      1 0 0 0 ···   [Y ] I     0 1 0 0 ···     0 0 1 0 ···   ......

kthcol

↓   0 ··· 0 0 0 0 ···   ......    ......  0k,k 0k,   N   [Y ]k[X]k =   =0 ··· 0 0 0 0 · · · ← kth row .         0 ,k I 0 ··· 0 1 0 0 ··· N       0 ··· 0 0 1 0 ···     0 ··· 0 0 0 1 ···   ......

Some particular examples of the matrix representation of Y mXn for different powers m, n are also given below.     0 0 0 0 0 0 0 ···   0 0 0 0 0 ···     0 0 0 1 0 0 0 ···     0 0 1 0 0 ···         0 0 0 0 1 0 0 ··· [Y ][X]2 = 0 0 0 1 0 ··· , [Y ][X]3 =   ,       0 0 0 0 0 1 0 ···     0 0 0 0 1 ···     0 0 0 0 0 0 1 ··· ......   ...... ......  ...... 118   0 0 0 0 0 0 0 ···       0 0 0 0 0 0 0 ··· 0 0 0 0 0 0 0 ···           0 0 0 0 0 0 0 ··· 0 0 0 0 0 0 0 ···         0 0 0 1 0 0 0 ··· 0 0 0 0 0 0 0 ···         [Y ]2[X]3 = 0 0 0 0 1 0 0 ··· , [Y ]4[X]3 = 0 0 0 1 0 0 0 ··· .             0 0 0 0 0 1 0 ··· 0 0 0 0 1 0 0 ···         0 0 0 0 0 0 1 ··· 0 0 0 0 0 1 0 ···     ......    ...... 0 0 0 0 0 0 1 ···   ......  ...... As we can notice in all these examples, the power of [X] rules the number of zero columns from the left to the right, while the power of [Y ] establishes the number of zero rows from the top to the bottom.

Let Ω = {(m, n):(m, n) ∈ Z2, m, n ≥ 0} and

G = {[Y ]m[X]n :(m, n) ∈ Ω}.

Then Lemma 5.3 implies that G is a monoid with usual multiplication of matrices. In addition, from the descriptions of the matrices [Y ]m[X]n given above we have that the function µ :Ω → G given by (m, n) 7→ [Y ]m[X]n is a bijection. Thus, in

order to specify an element in G, we need only to give (m, n) ∈ Z2. On the other hand, since each element in G is a row column finite matrix, then G ⊂ RCFM(R) ⊂ CFM(R). More importantly, by Lemma 5.3, G is a submoid of the multiplicative monoids RCFM(R) and CFM(R). Hence there are monoid inclusion maps ι : G → RCFM(R) and κ : G → CFM(R). On the other hand, let ϕ : R → RCFM(R) be the map defined as r 7→ rI, where I ∈ RCFM(R) is the identity matrix. Then ϕ is a ring homormorphism, which is in fact a monomorphism. Therefore, by Lemma 5.2, there is a unique ring 119

homomorphism Φ: R[G] → RCFM(R)

such that the following diagrams commute R[G] R[G] α Φ β Φ

ϕ R RCFM(R) G ι RCFM(R) Thus, for any X a = r(m,n)e(m,n) ∈ R[G], (m,n)∈Ω

where e(m,n) = e[Y ]m[X]n , we have X X Φ(a) = Φ(r(m,n)e(m,n)) = Φ((r(m,n)e(0,0))(e(m,n))) (m,n)∈Ω (m,n)∈Ω X = Φ(r(m,n)e(0,0))Φ(e(m,n)). (m,n)∈Ω Using the commutativity of the diagrams above, we have that

Φ(r(m,n)e(0,0)) = Φ(α(r(m,n))) = r(m,n)I, and

m n m n m n Φ(e(m,n)) = Φ(β([Y ] [X] )) = ι([Y ] [X] ) = [Y ] [X] .

Therefore,

X m n X m n Φ(a) = (r(m,n)I)([Y ] [X] ) = r(m,n)[Y ] [X] . (m,n)∈Ω (m,n)∈Ω

By definition of the monoid ring R[G], r(m,n) = 0 for all but a finite number of indexes (m, n) ∈ Ω. This implies that every element a in the monoid R[G] is mapped by Φ to a finite R-linear combination of elements of G ⊆ RCFM(R) ⊆ CFM(R), say

m1 n1 mk nk Φ(a) = r1[Y ] [X] + ··· + rk[Y ] [X] , for some ri ∈ R and (mi, ni) ∈ Ω, 1 ≤ i ≤ k. 120

Proposition 5.4. Let G = {[Y ]m[X]n :(m, n) ∈ Ω}. Then G is a linearly independent subset of RCFM(R). Consequently, Φ is a monomorphism.

Proof. We prove by mathematical induction that any finite subset of G is linearly independent. To this end, we use the description of the matrices [Y ]m[X]n given above. First, we note that the singleton {[Y ]m[X]n} is a linearly independent subset of RCFM(M) because each entry of [Y ]m[X]n is either zero or one. Assume now that any subset of G with k ≥ 1 elements is linearly independent.

Let (m1, n1),..., (mk, nk), (mk+1, nk+1) ∈ Ω be all distinct. Suppose that

m1 n1 mk nk mk+1 nk+1 M = r1[Y ] [X] + ··· + rk[Y ] [X] + rk+1[Y ] [X] = 0.

Without loss of generality, assume that mk+1 is the minimum power of [Y ]. Then mk+1 6= mi for all 1 ≤ i ≤ k. Thus the (mk+1 + 1, nk+1 + 1)-entry of

mk+1 nk+1 rk+1[Y ] [X] is equal to rk+1, while the same entry of all other terms of M are zero. It follows that the (mk+1 + 1, nk+1)-entry of M is rk+1, whence rk+1 = 0. Thus,

m1 n1 mk nk M = r1[Y ] [X] + ··· + rk[Y ] [X] = 0.

By induction hypothesis, the set {[Y ]m1 [X]n1 ,...,Y ]mk [X]nk } is linearly independent, from where we conclude that r1 = ··· = rk = 0.

In view of Proposition 5.4, the ring RCFM(R) contains an isomorphic copy of the group ring R[G]. In fact,

( k ) ∼ X mi ni R[G] = ri[Y ] [X] : k ∈ N, 1 ≤ i ≤ k ⊆ RCFM(R). i=1 Therefore, we can say that R[G] is a subring of the ring of row and column finite matrices RCFM(R), and so a subring of CFM(R). Our next goal is to present a description of the elements of R[G]. 121

First, since the matrices [Y ]m, [X]m and [Y ]m[X]m are in G for all m ≥ 0, then the matrices I − [Y ]k[X]k and

j−1 i−1 Eij = [Y ] (I − [Y ][X])[X] are in R[G]. Using the description of the matrix [Y ]k[X]k previously given, we deduce that a description of the matrix I − [Y ]k[X]k is

kthcol

↓   1 ··· 0 0 0 0 ···   ......    ......  I 0   k,k k,N   I − [Y ]k[X]k =   =0 ··· 1 0 0 0 · · · ← kth row         0 ,k 0 , 0 ··· 0 0 0 0 ··· N N N       0 ··· 0 0 0 0 ···     0 ··· 0 0 0 0 ···   ......

On the other hand, observe that Eij is the matrix representation of the

endomorphism ebij introduced in §2.1. Since ebij(ei) = ej and ebij(ek) = 0 for k 6= i, we

have that Eij is the matrix whose all entries are zero except in the (i, j)-entry where

it is 1. The matrix Eij is called matrix unit. In consequence, the ring R[G] contains all finite combinations of these matrix units. In particular, it contains all matrices of the form 122

nthcol

↓   a11 ··· a1n 0 0 0 ···    ......     ......    A 0m,N   th   =am1 ··· amn 0 0 0 · · · ← m row .       0N,n 0N,N  0 ··· 0 0 0 0 ···        0 ··· 0 0 0 0 ···      0 ··· 0 0 0 0 ···   ......

We can take R-linear combinations of the previous matrices with matrices of the form [Y ]k[X]k to obtain new matrices that belong to R[G]. For example, for all r ∈ R, the following matrices belong to R[G]:   a11 ··· a1n 0 0 0 ···    ......   ......        a ··· a 0 0 0 ···  n1 nn  A 0k,    N  k k     + r[Y ] [X] =  0 ··· 0 r 0 0 ··· .   0N,k 0N,N    0 ··· 0 0 r 0 ···      0 ··· 0 0 0 r ···    ......  ......

These are of particular interest since the set of all such matrices constitute what is called an invertible algreba in [95]. Because of their particular shape, the nickname kite matrices have been given to them in the same paper. Note that the top-left corner of a kite matrix is a square matrix, and that these matrices have only one infinite tail. However, there is no reason to focus on only those matrices, because the ring R[G] also contains non-square kite matrices with finitely many 123

infinite tails. For instance, the following matrix belongs to R[G]:

nthcol

↓   a11 ··· a1n 0 0 0 ···    ......   ......      th am1 ··· amn t 0 0 · · · ← m row .      0 ··· s r t 0 ···        0 ··· 0 s r t ···      0 ··· 0 0 s r ···   ......

The shape of the last matrices along the fact that all matrix units Eij belong to R[G] suggest that the inclusion R[G] ⊆ RCFM(R) is proper, as the next example shows.

Example 5.5. Let M be the matrix that has a 1 in the (i, 2i)-entry and zero everywhere else. Then M ∈ RCFM(R) but M/∈ R[G]. To prove this last statement, recall that a matrix A ∈ R[G] can be written uniquely as

m1 n1 mk nk A = r1[Y ] [X] + ··· + rk[Y ] [X] ,

for some ri ∈ R and (mi, ni) ∈ Ω. Let m = max{mi : 1 ≤ i ≤ k} and

n = max{ni : 1 ≤ i ≤ k}. In addition, remember that the powers mi of [Y ] rules the number of zero rows from the top to the bottom, and the powers ni of [X] establish the number of zero columns from the left to the right in the matrix [Y ]mi [X]ni . Thus the nonzero entries of A lie within the nonzero diagonals of the matrix [Y ]m + [X]n. This property is clearly not satisfied by the matrix M, so M cannot be in R[G]. 124

5.4 Reflexive Rings and Column Finite Matrix Rings

A very important topic in the theory of column finite matrices and row column finite matrices is to relate properties of the ring R and the matrix rings RCFM(R) and CFM(R) (see [92] for more details). In this section we determine whether the rings R[G], RCFM(R) and CFM(R) are reflexive provided that the ring R is reflexive.

For any matrix A ∈ CFM(R) and any matrix unit Eii ∈ R[G], the product

EiiA yields a matrix whose ith row is equal to the ith row of A, and 0 everywhere else. Thus, the matrix EiiA ∈ CFM(R) but not necessarily EiiA ∈ RCFM(A), unless the ith row of A has finitely many nonzero entries.. Similarly, the matrix

AEjj has jth column equal to the jth column of A, and 0 everywhere else. In contrast to the first case, since A is a column finite matrix, we always have

AEjj ∈ RCFM(R). Multiplying on both sides the matrix A by matrix units we get the matrix

EiiAEjj whose entries are all zero except the (i, j)-entry, where it is equal to the

(i, j)-entry of A. Thus, if A = (akl) ∈ CFM(R), then

EiiAEjj = aijEij = Eijaij.

Recall that a ring R is called reflexive if for all a, b ∈ R we have that aRb = 0 implies bRa = 0. This class of rings is clearly closed under direct sums, direct products, but it is neither closed under subrings and homomorphic images. For more details see [45, Examples 2.7 and 2.8]. The next result is the main theorem of this chapter.

Theorem 5.6. Let R be a ring. 1. CFM(R) is reflexive if and only if R is reflexive.

2. RCFM(R) is reflexive if and only if R is reflexive. 125

3. R[G] is reflexive if and only if so is R.

Proof. 1. Let A = (aij),B = (bij) column finite matrices, and assume that

AMB = 0 for all M ∈ CFM(R). Then for all i, j ∈ N we have that

A(EiiMEjj)B = 0, so for all k, l ∈ N we get

Ekk (AEiiMEjjB) Ell = 0

Using the associativity of the product of matrices and the fact that Err is idempotent, we obtain

(EkkAEii)(EiiMEjj)(EjjBEll) = 0,

or equivalently,

0 = (Ekiaki)(Eijmij)(Ejlbjl) = Ekl(akimijbjl).

Since M ∈ S was arbitrary, we conclude

akiRbjl = 0 ∀ i, j, k, l ∈ N.

Since R is a reflexive ring, then

bjlRaki = 0 ∀ i, j, k, l ∈ N. (5.4.1)

We now prove that BMA = 0 for all M ∈ CFM(R). Assume, for the sake

of contradiction, there exists a matrix C = (cij) ∈ CFM(R) such that

BCA 6= 0. Then there exists k, l ∈ N such that the (k, l)-entry of BCA is nonzero, i.e,

Ekk(BCA)Ell 6= 0 ⇔ (EkkBC)(AEll) 6= 0.

Note that AEll is the matrix whose lth column is the lth column of A, and 0 everywhere else. Since A is a column finite matrix, then there exists a finite 126 nonempty set Γ ⊂ N such that the (l, λ)-entry of A is nonzero for each λ ∈ Γ. Thus, X AEll = alλElλ. λ∈Γ

On the other hand, EkkBC is the matrix whose kth row is equal to the kth row of BC, and 0 everywhere else. Since B and C are column finite matrices, then the kth row of BC could have infinitely many nonzero entries. However, since AEll is column finite, when computing the product of the kth row of BC with the lth column of A, we need to consider only the (k, γ)-entries of BC, where γ ∈ Γ. Thus, ! ! X X (EkkBC)(AEll) = (EkkBC)Eγγ alλElλ . γ∈Γ λ∈Γ Clearly, X X (EkkBC)Eγγ = EkkB CEγγ. γ∈Γ γ∈Γ P The matrix γ∈Γ CEγγ has all columns equal to zero, whith the probably exception of the γ columns, where γ ∈ Γ. Since C is a column finite matrix and Γ is a finite set, there exists a finite set ∆ ⊂ N such that

X X X CEγγ = EδδCEγγ. γ∈Γ γ∈Γ δ∈∆ Thus,

X X X X X EkkB CEγγ = EkkB EδδCEγγ = EkkBEδδCEγγ γ∈Γ γ∈Γ δ∈∆ γ∈Γ δ∈∆ X X = (EkkBEδδ)(EδδCEγγ) γ∈Γ δ∈∆ X X = (bkδEkδ)(cδγEδγ) γ∈Γ δ∈∆ X X = bkδcδγEkγ. γ∈Γ δ∈∆ 127

Therefore, ! ! X X X (EkkBC)(AEll) = bkδcδγEkγ alλElλ γ∈Γ δ∈∆ λ∈Γ X X X = bkδcδγalλEkγElλ γ∈Γ δ∈∆ λ∈Γ

By (5.4.1), for any choice of k, δ, l, λ ∈ N we have bkδRalλ = 0. Thus,

EkkBCAEll = (EkkBC)(AEll) = 0.

This implies that the (k, l)-entry of the matrix BCA is zero (a contradiction). Therefore, BCA = 0 and so, the ring CFM(R) is reflexive. To prove the converse, notice that if S is a reflexive ring and e is an idempotent in S, then the ring eSe is also reflexive. Thus, if S = CFM(R) ∼ is a reflexive ring, then E11SE11 is a reflexive ring. But E11SE11 = R.

2. The proof of this statement is similar to the proof of (1). We only have to replace M ∈ CFM(R) by a matrix M ∈ RCFM(R). For the converse,

observe that the matrix unit E11 ∈ RCFM(R).

3. The matrix unit Eij belongs to the monoid ring R[G], so the proof of this statement is similar to the proof of (1).

Corollary 5.7. Let R be a reflexive ring. The rings R[G], RCFM(R), and CFM(R) are reflexive non-Dedekind-finite.

Proof. Since XY = 1M but YX 6= 1M , then we have [X][Y ] = I but [Y ][X] 6= I, where I ∈ RCFM(R) is the identity matrix. This implies that R[G], RCFM(R) and CFM(R) are non-Dedekind finite rings.

N Corollary 5.8. Let R be a ring and M = R . The ring EndR(M) of left endomorphisms of M is a reflexive non-Dedekind finite ring if and only if R is reflexive. 128 ∼ Proof. It follows from the fact that EndR(M) = CFM(R).

Corollary 5.9. Let V a vector space of countable infinite dimension over a division

ring D. Then the ring of endomorphisms EndD(V ) is a non-Dedekind finite reflexive ring.

Proof. Clearly any division ring D is a reflexive ring. On the other hand, V ∼= DN, where the result follows.

Corollary 5.10. Let F be a field. The monoid algebra F [G], and the F -algebas CFM(F ) and RCFM(F ) are non-Dedekind finite reflexive rings. 129 References

[1] V. Camillo and P. P. Nielsen, “McCoy rings and zero-divisors,” J. Pure Appl. Algebra, vol. 212, no. 3, pp. 599–615, 2008.

[2] G. Marks, “A taxonomy of 2-primal rings,” J. Algebra, vol. 266, no. 2, pp. 494–520, 2003.

[3] P. P. Nielsen, “Semi-commutativity and the McCoy condition,” J. Algebra, vol. 298, no. 1, pp. 134–141, 2006.

[4] M. B. Rege and S. Chhawchharia, “Armendariz rings,” Proc. Japan Acad. Ser. A Math. Sci., vol. 73, no. 1, pp. 14–17, 1997.

[5] S. Szabo, “Minimal reversible nonsymmetric rings,” J. Pure Appl. Algebra, vol. 223, no. 11, pp. 4583–4591, 2019.

[6] S. Szabo, “Some minimal rings related to 2-primal rings,” Comm. Algebra, vol. 47, no. 3, pp. 1287–1298, 2019.

[7] B. Corbas and G. Williams, “Rings of order p5 Part I. Nonlocal rings,” Journal of Algebra, vol. 231, no. 2, pp. 677 – 690, 2000.

[8] B. Corbas and G. Williams, “Rings of order p5 Part II. local rings,” Journal of Algebra, vol. 231, no. 2, pp. 691 – 704, 2000.

[9] G. Birkenmeier, H. Heatherly, and E. Lee, Completely Prime Ideals and Radicals in Near-Rings, pp. 63–73. Dordrecht: Springer Netherlands, 1995.

[10] W. K. Nicholson and J. F. Watters, “Normal radicals and normal classes of rings,” Journal of Algebra, vol. 59, pp. 5–15, 1979.

[11] M. Hongan, “Some remarks on normal classes of semiprime rings,” Mathematical Journal of Okayama University, vol. 25, no. 2, pp. 139–143, 1983.

[12] J. Baeck, N. K. Kim, Y. Lee, and P. P. Nielsen, “Zero-divisor placement, a condition of Camillo, and the McCoy property,” J. Pure Appl. Algebra, vol. 224, no. 12, pp. 106432, 14, 2020.

[13] J. Chen, Z. Lei, and Z. Ying, “A question on McCoy rings,” Bull. Austral. Math. Soc., vol. 76, no. 1, pp. 137–141, 2007.

[14] Y. Hirano, “On annihilator ideals of a polynomial ring over a noncommutative ring,” J. Pure Appl. Algebra, vol. 168, no. 1, pp. 45–52, 2002.

[15] C. Y. Hong, Y. C. Jeon, N. K. Kim, and Y. Lee, “The McCoy condition on noncommutative rings,” Comm. Algebra, vol. 39, no. 5, pp. 1809–1825, 2011. 130

[16] N. K. Kim and Y. Lee, “On a ring property unifying reversible and right duo rings,” J. Korean Math. Soc., vol. 50, no. 5, pp. 1083–1103, 2013.

[17] N. K. Kim and Y. Lee, “Extensions of reversible rings,” J. Pure Appl. Algebra, vol. 185, no. 1-3, pp. 207–223, 2003.

[18] C. Huh, Y. Lee, and A. Smoktunowicz, “Armendariz rings and semicommutative rings,” Comm. Algebra, vol. 30, no. 2, pp. 751–761, 2002.

[19] H. Chimal-Dzul and S. Szabo, “Minimal reflexive nonsemicommutative rings,” Journal of Algebra and its Applications, 2021. https://doi.org/10.1142/S0219498822500840.

[20] H. Chimal-Dzul, S. L´opez-Permouth, and S. Szabo, “Characterization of certain Morita context rings,” Pre-print. Submitted.

[21] H. Chimal-Dzul, S. L´opez-Permouth, and S. Szabo, “Abelian right mccoy rings and related notions,” Pre-print. Submitted.

[22] J. Lambek, “On the representation of modules by sheaves of factor modules,” Canad. Math. Bull., vol. 14, pp. 359–368, 1971.

[23] J. Lambek, “On the representation of modules by sheaves of modules of quotients,” in Ring Theory (R. GORDON, ed.), pp. 231 – 234, Academic Press, 1972.

[24] D. D. Anderson and V. Camillo, “Semigroups and rings whose zero products commute,” Comm. Algebra, vol. 27, no. 6, pp. 2847–2852, 1999.

[25] T. P. Calci, S. Halicioglu, and A. Harmanci, “Symmetric property of rings with respect to the Jacobson radical,” Commun. Korean Math. Soc., vol. 34, no. 1, pp. 43–54, 2019.

[26] A. Harmanci, H. Kose, and B. Ungor, “On weak symmetric property of rings,” Southeast Asian Bull. Math., vol. 42, no. 1, pp. 31–40, 2018.

[27] C. Huh, H. K. Kim, N. K. Kim, and Y. Lee, “Basic examples and extensions of symmetric rings,” J. Pure Appl. Algebra, vol. 202, no. 1-3, pp. 154–167, 2005.

[28] H. K. Kim, T. K. Kwak, S. I. Lee, Y. Lee, S. J. Ryu, H. J. Sung, and S. J. Yun, “A generalization of symmetric ring property,” Bull. Korean Math. Soc., vol. 53, no. 5, pp. 1309–1325, 2016.

[29] D. W. Jung, T. K. Kwak, M. J. Lee, and Y. Lee, “Ring properties related to symmetric rings,” Internat. J. Algebra Comput., vol. 24, no. 7, pp. 935–967, 2014. 131

[30] G. Marks, “Reversible and symmetric rings,” J. Pure Appl. Algebra, vol. 174, no. 3, pp. 311–318, 2002.

[31] T. Subedi and D. Roy, “On a common generalization of symmetric rings and quasi duo rings,” Algebra Discrete Math., vol. 29, no. 2, pp. 249–258, 2020.

[32] P. M. Cohn, “Reversible rings,” Bull. London Math. Soc., vol. 31, no. 6, pp. 641–648, 1999.

[33] B.-O. Kim and Y. Lee, “Minimal noncommutative reversible and reflexive rings,” Bull. Korean Math. Soc., vol. 48, no. 3, pp. 611–616, 2011.

[34] J. Huang, H.-l. Jin, Y. Lee, and Z. Piao, “Reversible and pseudo-reversible rings,” Bull. Korean Math. Soc., vol. 56, no. 5, pp. 1257–1272, 2019.

[35] D. W. Jung, N. K. Kim, Y. Lee, and S. J. Ryu, “On properties related to reversible rings,” Bull. Korean Math. Soc., vol. 52, no. 1, pp. 247–261, 2015.

[36] H. E. Bell, “Near-rings in which each element is a power of itself,” Bull. Austral. Math. Soc., vol. 2, pp. 363–368, 1970.

[37] G. Shin, “Prime ideals and sheaf representation of a pseudo symmetric ring,” Trans. Amer. Math. Soc., vol. 184, pp. 43–60, 1973.

[38] L. Xu and W. Xue, “Structure of minimal non-commutative zero-insertive rings,” Math. J. Okayama Univ., vol. 40, pp. 69–76 (2000), 1998.

[39] L. Motais de Narbonne, “Anneaux semi-commutatifs et unis´eriels;anneaux dont les id´eauxprincipaux sont idempotents,” in Proceedings of the 106th National Congress of Learned Societies (Perpignan, 1981), pp. 71–73, Bib. Nat., Paris, 1982.

[40] K.-Y. Ham, Y. C. Jeon, J. Kang, N. K. Kim, W. Lee, Y. Lee, S. J. Ryu, and H.-H. Yang, “IFP rings and near-IFP rings,” J. Korean Math. Soc., vol. 45, no. 3, pp. 727–740, 2008.

[41] W. Chen, “On π-semicommutative rings,” J. Math. Res. Appl., vol. 36, no. 4, pp. 423–431, 2016.

[42] L. Wang and J. Wei, “Weakly semicommutative rings and strongly regular rings,” Kyungpook Math. J., vol. 54, no. 1, pp. 65–72, 2014.

[43] G. Mason, “Reflexive ideals,” Comm. Algebra, vol. 9, no. 17, pp. 1709–1724, 1981.

[44] L. Zhao, X. Zhu, and Q. Gu, “Reflexive rings and their extensions,” Mathematica Slovaca, vol. 63, no. 3, 01 Jun. 2013. 132

[45] T. K. Kwak and Y. Lee, “Reflexive property of rings,” Comm. Algebra, vol. 40, no. 4, pp. 1576–1594, 2012.

[46] K. R. F. a. Frank W. Anderson, Rings and Categories of Modules. Graduate Texts in Mathematics 13, Springer New York, 2nd ed., 1974.

[47] E. H. Feller, “Properties of primary noncommutative rings,” Transactions of the American Mathematical Society, vol. 89, no. 1, pp. 79–91, 1958.

[48] Y. Hirano, C.-Y. Hong, J.-Y. Kim, and J. K. Park, “On strongly bounded rings and duo rings,” Comm. Algebra, vol. 23, no. 6, pp. 2199–2214, 1995.

[49] W. Xue, “On strongly right bounded finite rings,” Bull. Austral. Math. Soc., vol. 44, no. 3, pp. 353–355, 1991.

[50] W. Xue, “Structure of minimal noncommutative duo rings and minimal strongly bounded nonduo rings,” Comm. Algebra, vol. 20, no. 9, pp. 2777–2788, 1992.

[51] V. R. Chandran, “On duo-rings,” Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8), vol. 58, no. 6, pp. 823–827, 1975.

[52] J. M. Habeb, “A note on zero commutative and duo rings,” Math. J. Okayama Univ., vol. 32, pp. 73–76, 1990.

[53] G. Marks, “Duo rings and Ore extensions,” J. Algebra, vol. 280, no. 2, pp. 463–471, 2004.

[54] G. Thierrin, “On duo rings,” Canadian Mathematical Bulletin, vol. 3, no. 2, p. 167–172, 1960.

[55] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.

[56] T. Y. Lam, A first course in noncommutative rings, vol. 131 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2 ed., 2001.

[57] H. Chimal-Dzul and S. Szabo, “Classification of nonsemicommutative rings,” Pre-print. Submitted.

[58] G. F. Birkenmeier, H. E. Heatherly, and E. K. Lee, “Completely prime ideals and associated radicals,” in Ring theory (Granville, OH, 1992), pp. 102–129, World Sci. Publ., River Edge, NJ, 1993.

[59] S. U. Hwang, Y. C. Jeon, and Y. Lee, “Structure and topological conditions of NI rings,” J. Algebra, vol. 302, no. 1, pp. 186–199, 2006. 133

[60] G. Marks, “Direct product and power series formations over 2-primal rings,” in Advances in ring theory (Granville, OH, 1996) (S. K. Jain and S. T. Rizvi, eds.), Trends Math., pp. 239–245, Boston, MA: Birkh¨auserBoston, 1997. [61] Y. Lee, C. Huh, and H. K. Kim, “Questions on 2-primal rings,” Comm. Algebra, vol. 26, no. 2, pp. 595–600, 1998. [62] G. Marks, “On 2-primal Ore extensions,” Comm. Algebra, vol. 29, no. 5, pp. 2113–2123, 2001. [63] A. R. Nasr-Isfahani, “Ore extensions of 2-primal rings,” J. Algebra Appl., vol. 13, no. 3, pp. 1350117, 7, 2014. [64] W.-x. Chen and S.-y. Cui, “On weakly semicommutative rings,” Commun. Math. Res., vol. 27, no. 2, pp. 179–192, 2011. [65] W. Chen, “On constant products of elements in skew polynomial rings,” Bull. Iranian Math. Soc., vol. 41, no. 2, pp. 453–462, 2015. [66] E. Hashemi, M. Hamidizadeh, and A. Alhevaz, “Some types of ring elements in Ore extensions over noncommutative rings,” J. Algebra Appl., vol. 16, no. 11, pp. 1750201, 17, 2017. [67] M. Jiang, Y. Wang, and Y. Ren, “Extensions and topological conditions of NJ rings,” Turkish J. Math., vol. 43, no. 1, pp. 44–62, 2019. [68] Y. Wang, M.-m. Jiang, and Y.-l. Ren, “Ore extensions over weakly 2-primal rings,” Commun. Math. Res., vol. 32, no. 1, pp. 70–82, 2016. [69] Y. Wang, M. Jiang, and Y. Ren, “π-Armendariz rings relative to a monoid,” Front. Math. China, vol. 11, no. 4, pp. 1017–1036, 2016. [70] Y. Chun, Y. C. Jeon, S. Kang, K. N. Lee, and Y. Lee, “A concept unifying the Armendariz and NI conditions,” Bull. Korean Math. Soc., vol. 48, no. 1, pp. 115–127, 2011. [71] J. Han, Y. Lee, and S. P. Yang, “Rings over which polynomial rings are NI,” in Contemporary ring theory 2011, pp. 1–9, World Sci. Publ., Hackensack, NJ, 2012. [72] A. R. Nasr-Isfahani, “On NI skew polynomial rings,” Comm. Algebra, vol. 43, no. 12, pp. 5113–5120, 2015. [73] C. Selvaraj and S. Petchimuthu, “NI rings which are weakly π-regular,” Int. Electron. J. Algebra, vol. 6, pp. 31–37, 2009. [74] C. Selvaraj and S. Petchimuthu, “Clean and exchange conditions of NI rings,” Bull. Pure Appl. Math., vol. 5, no. 1, pp. 91–100, 2011. 134

[75] A. Tehranchi and K. Paykan, “A characterization of 2-primal and NI rings over skew inverse Laurent series rings,” J. Algebra Appl., vol. 18, no. 12, pp. 1950221, 16, 2019. [76] O. D. Artemovych, “Associative rings in which nilpotents form an ideal,” Studia Sci. Math. Hungar., vol. 56, no. 2, pp. 177–184, 2019. [77] W. Chen, “Polynomial rings over weak Armendariz rings need not be weak Armendariz,” Comm. Algebra, vol. 42, no. 6, pp. 2528–2532, 2014. [78] W. Chen, “On linearly weak Armendariz rings,” J. Pure Appl. Algebra, vol. 219, no. 4, pp. 1122–1130, 2015. [79] E. Mart´ınez-Moroand S. Szabo, “On codes over local Frobenius non-chain rings of order 16,” in Noncommutative rings and their applications, vol. 634 of Contemp. Math., pp. 227–241, Amer. Math. Soc., Providence, RI, 2015. [80] J. B. Derr, G. F. Orr, and P. S. Peck, “Noncommutative rings of order p4,” J. Pure Appl. Algebra, vol. 97, no. 2, pp. 109–116, 1994. [81] B. R. McDonald, Finite Rings With Identity. Pure and Applied Mathematics, Marcel Dekker Inc, 1974. [82] W. E. Clark and D. A. Drake, “Finite chain rings,” Abhandlungen aus dem Mathematischen Seminar der Universit¨atHamburg, vol. 39, pp. 147–153, Sep 1973. [83] S. A. Amitsur, “Rings of quotients and morita contexts,” Journal of Algebra, vol. 17, no. 2, pp. 273 – 298, 1971. [84] H. Bass, The Morita Theorems. Mimeographed Notes, University of Oregon, 1962. [85] P. Loustaunau and J. Shapiro, “Morita contexts,” in Non-Commutative Ring Theory (S. K. Jain and S. R. L´opez-Permouth, eds.), (Berlin, Heidelberg), pp. 80–92, Springer Berlin Heidelberg, 1990. [86] A. Sands, “Radicals and morita contexts,” Journal of Algebra, vol. 24, no. 2, pp. 335 – 345, 1973. [87] G. Tang, C. Li, and Y. Zhou, “Study of morita contexts,” Communications in Algebra, vol. 42, no. 4, pp. 1668–1681, 2014. [88] T. Y. Lam, Lectures on Modules and Rings, vol. 189 of Graduate Texts in Mathematics. Springer-Verlag New York, 1 ed., 1999. [89] N. H. McCoy, “Remarks on divisors of zero,” Amer. Math. Monthly, vol. 49, pp. 286–295, 1942. 135

[90] E. P. Armendariz, “A note on extensions of Baer and P.P.-rings,” J. Austral. Math. Soc., vol. 18, pp. 470–473, 1974.

[91] D. Anderson and V. Camillo, “Armendariz rings and gaussian rings,” Communications in Algebra, vol. 26, no. 7, pp. 2265–2272, 1998.

[92] V. Camillo, F. Costa-Cano, and J. Sim´on,“Relating properties of a ring and its ring of row and column finite matrices,” Journal of Algebra, vol. 244, no. 2, pp. 435 – 449, 2001.

[93] P. P. Nielsen, “Row and column finite matrices,” Proceedings of the American Mathematical Society, vol. 135, no. 9, pp. 2689–2697, 2007.

[94] C. P. Milies and S. K. Sehgal, An Introduction to Group Rings. Algebra and Applications, Kluwer, 2002 ed., 2002.

[95] S. R. L´opez-Permouth, J. Moore, N. Pilewski, and S. Szabo, “Algebras having bases that consist solely of units,” Israel Journal of Mathematics, vol. 208, pp. 461–482, Sep 2015. Alphabetical Index

(S I) ring, 22 endomorphism ring, 114

2-primal ring, 31 faithful bimodule homomorphism, 62, abelian, 91 64 abelian ring, 36 formal triangular matrix ring, 61

Armendariz ring, 84 group algebra, 51 Armendarizlike ring, 85 idempotent, 16 Baer’s lower nilradical, 29 IFP ring, 22 Baer-McCoy radical, 29 indecomposable ring, 22 index of nilpotency, 41 central element, 36 index of the ring, 36 column finite matrix, 110 Insertion-of-Factors-Property, 22 Commutative ring

nilradical, 28 Jacobson radical, 30 completely prime ideal, 33 K¨othe’sconjecture, 62, 78 completely semiprime, 33 kite matrices, 122 continuous endomorphism, 110 left annihilator, 22 CZ2 left chain ring, 42 see reversible ring, 20 left duo ring, 19, 26 Dedekind-finite ring, 36 left McCoy ring, 84

Dihedral group D8, 52 left real McCoy ring, 87 Directly-finite, 36 left zero-divisor, 16 domain, 18 Levitzki radical, 30 duo ring, 26 local ring, 22 136 137 locally nilpotent ideal, 29 reflexive ideal, 24

reflexive ring, 19, 63 matrix unit, 121 reversible ring, 18, 20 McCoy ring, 84 right annihilator, 22 minimal ring, 11, 21 right Armendarizlike ring, 85 Morita context, 61 right chain ring, 42 Morita context reflexive, 64 right duo ring, 19, 26 Morita context ring with zero trace right McCoy ring, 84 ideals, 61 right real McCoy ring, 86, 87 Morita context rings, 61 right zero-divisor, 16 NI, 35 Ring NI ring, 31 reflexive, 24 nil ideal, 29 ring of continuous endomorphisms, 111 nilpotent, 16 row and column finite matrix, 110 nilpotent ideal, 29 nilpotent ring, 29 semicommutative ring, 19, 22 non-squere kate matrix, 122 semiprime ideal, 29 NR ring, 35 semiprime ring, 24, 63 strongly left McCoy ring, 85 polynomial semicommutative ring, 85 strongly McCoy ring, 85 prime ideal, 29 strongly right McCoy ring, 85 prime radical, 29 symmetric ring, 18, 19 prime ring, 63

PS I ring, 31 trace ideals, 61 pseudo symmetric ring, 32 upper nilradical, 30 real McCoy ring, 87 reduced, 18 weakly 2-primal ring, 31 138

ZC3 ring, see symmetric ring zero-insertive ring, 22 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

Thesis and Dissertation Services ! !