L-Stability in Rings and Left Quasi-Duo Rings

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2018-05-28 L-stability in Rings and Left Quasi-duo Rings

Horoub, Ayman Mohammad Abedalqader

Horoub, A. M. A, (2018). L-stability in Rings and Left Quasi-duo Rings (Unpublished doctoral thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/31958 http://hdl.handle.net/1880/106702 doctoral thesis

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L-stability in Rings and Left Quasi-duo Rings

Ayman Mohammad Abedalqader Horoub

A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

GRADUATE PROGRAM IN MATHEMATICS AND STATISTICS

CALGARY, ALBERTA May, 2018

c Ayman Mohammad Abedalqader Horoub 2018 Abstract

A ring R is said to have stable range 1 if, for any element a ∈ R and any left ideal L of R,

Ra + L = R implies a − u ∈ L for some unit u in R. Here we insist only that this holds for all L in some non-empty set L(R) of left ideals of R, and say that R is left L-stable in this case. We say that a class C of rings is affordable if C is the class of left L-stable rings for some left idealtor L. In addition to the rings of stable range 1, it is known that the left uniquely generated rings and the rings with internal cancellation are both affordable. Here, we explore L-stability in general, derive some properties of this phenomenon and show that it captures many well-known results. This in turn yields new information about the left uniquely generated rings and the internally cancellable rings, and enables us to answer some open questions related to them. More importantly, we show that the directly finite rings are affordable, which gives a new perspective on these rings and the plethora of open questions related to them.

Next, we turn to the class of left quasi-duo rings, that is, rings R in which every maximal left ideal is an ideal. But, surprisingly, these rings have many nice natural characterizations and properties that have passed unnoticed since they were introduced in 1995. Here we discuss this class of rings and prove some interesting new results for them. In particular, we introduce the notion of left width for rings, and use it to give a characterization of any left quasi-duo ring that has a ﬁnite left width. After that, a characterization of the I-ﬁnite left quasi-duo rings will be given.

Finally, we introduce the left-max idealtor which is deﬁned in terms of the maximal left ideals for any ring R. Then we study and characterize its associated rings which will be called the left- max stable rings. We also show that the class of left quasi-duo rings is neither left-max stable nor aﬀordable.

ii Acknowledgement

I acknowledge with deep gratitude and appreciation my supervisor Professor W.K. Nicholson for his continuous support of my Ph.D study and related research and for his patience and motivation. His guidance helped me throughout my researching and writing of this thesis. I could not have imagined having a better supervisor for my Ph.D study.

Especial thanks to Professor Berndt Brenken, Professor Michael Jacobson and Professor Mo- hamed Yousif from my Exam Committee for their valuable feedback and good comments.

Grateful thanks to the University of Calgary for its remarkable facilities and for granting me the opportunity to pursue my Ph.D study. And many thanks to all people who helped me throughout my study in this University, especially, Professor Antony Ware and Yanmei Fei for their help.

Finally, great thanks to my family; my parents, brothers and sisters for their continuous guidance and moral support throughout my academic life.

Thanks for all your encouragement!

iii To my parents, brothers and sisters for their encouragement and support.

To my most favorite topic, “Rings”.

iv Table of Contents

Page

Abstract ii

Acknowledgement iii

Dedication iv

Table of Contents v

List of Symbols and Abbreviations vii

I L-stability in Rings1

1 Introduction: Definitions, Examples and Fundamental Properties2 1.1 Importance of introducing the L-stability Notion ...... 2 1.2 Definitions and Four Major Examples ...... 3 1.3 Two Properties of Left Idealtors ...... 10 1.4 Basic Features and Essential Results for L-stability ...... 13 1.4.1 A Useful Tool to Check Ring Affordability ...... 14 1.4.2 DF Rings Revisited ...... 16 1.4.3 Special Left L-stable Elements ...... 17 1.4.4 A New Look at Three Old Well-Known Results ...... 19 1.4.5 An Extension of a Theorem of Bass ...... 22

2 Related Rings 24 2.1 Factor Rings ...... 24 2.2 Corner Rings ...... 27 2.3 Direct Products ...... 29 2.4 Subrings ...... 31 2.4.1 Motivation ...... 31 2.4.2 Ideal Extensions ...... 31 2.5 Polynomial Rings and Rings of Formal Power Series ...... 33 2.6 Matrix Rings ...... 35 2.6.1 L-stability is not a Morita Invariant Property of Rings in General ...... 36 2.6.2 Generalized Matrix Rings ...... 37 2.6.3 Generalized Upper Triangular Matrix Rings and Generalized Context-null Extensions ...... 39

v 3 Special Classes of Left Idealtors 43 3.1 Normal Left Idealtors ...... 43 3.2 Closed Left Idealtors ...... 44 3.3 Products of left L-stable elements ...... 46 3.4 Nice Left Idealtors ...... 50

II Left Quasi-duo Rings 54

4 Examples and Basic Results 55 4.1 Motivation ...... 55 4.2 Module Rings and a “Simple-Modules” Characterization of Left Quasi-duo Rings . . 57 4.3 Examples and Elementary Properties ...... 59 4.4 A “Very Semisimple-Modules” Characterization of Left Quasi-duo Rings ...... 64 4.5 Generalized Upper Triangular Matrix Ring over Left Quasi-duo Rings ...... 66

5 Advanced Results for Left Quasi-duo Rings 68 5.1 Left Quasi-duo Rings with Finite Left Width ...... 68 5.1.1 The Width Theorem and the Left Width of Rings ...... 68 5.1.2 The Width Theorem for Left Quasi-duo Rings ...... 73 5.2 Left Quasi-duo Rings that are I-finite ...... 76 5.2.1 Left Quasi-duo I-finite Rings that are Semiprime ...... 76 5.2.2 Characterization of Left Quasi-duo I-finite Rings ...... 78 5.3 Left-Max idealtor and Left-Max stable Rings: ...... 81

Bibliography 85

Index 94

vi List of Symbols and Abbreviations

⇒ implication ⇔ logical equivalence := or =: introducing definitions {x | P } the set of all x such that P (x) N set of natural numbers Z set of integers Q set of rational numbers R set of real numbers Zn integers modulo n a Z(p,q) Z(p,q) = { b ∈ Q | p - b, q - b} for primes p, q in Z nZ = Zn subgroup (ideal) of Z generated by n ∪· disjoint union of sets =∼ isomorphic to R\A difference set A∗ A\{0} |A| cardinality of set A (number of elements in A) α : A → B or A −→α B mapping α from A to B α(a) or aα used interchangeably for the image of a under mapping α a 7→ aα definition of mapping α ,→ inclusion mapping αβ or β ◦ α composite of mapping α and β (first α, then β ) 1A identity mapping on set A α−1 inverse of mapping α W (R); W (F ) Weyl algebra over R; over a field F R[x] ring of polynomials in x over R R[[x]] formal power series ring in x over R Mn(R) n × n matrices over a ring R Πi∈I Ri direct product of the rings {Ri} hrii elements in Πi∈I Ri RI direct product of I copies of ring R Rhxi | i ∈ Ii free ring over a ring R generated by {xi | i ∈ I} πk canonical projections πk :Πi∈I Ri → Rk Mn[Ri,Vij] generalized n × n matrix ring over the rings Ri [ri, vij] elements of Mn[Ri,Vij] Tn[Ri,Vij] generalized n × n upper triangular matrix ring over the rings Ri

vii T2[Ri,Vij] split-null extension (R, RVS, SWR, S) Morita context CNn[Ri,Vij] generalized n × n context-null extension hom(M,N) group of module homomorphisms end(M) endomorphism ring of left R-module M m¯ or m + N elements of the factor module M/N l(m) left annihilator of an element m in a left R-module M l(M) left annihilator of a left R-module M max N ⊆ RMN is a maximal submodule of a left R-module M ess N ⊆ RMN is an essential submodule of a left R-module M sm N ⊆ RMN is a small submodule of a left R-module M Rm principal left R-module generated by m S(M) or soc(M) socle of a left R-module M R/A factor ring r¯ or r + A elements of the factor ring R/A hai principal ideal generated by a A C RA is an ideal of a ring R A ⊆RRA is a left ideal of a ring R max A ⊆ RRA is a maximal left ideal of a ring R Sl(R) or S(RR) left socle of a ring R e idempotent element in a ring R, that is, e2 = e eRe corner ring with identity e2 = e a−1 multiplicative inverse of an element a in a ring R −a additive inverse of an element a in a ring R l(a) left annihilator of an element a in a ring R r(a) right annihilator of an element a in a ring R l(A) left annihilator of a left ideal A in a ring R r(A) right annihilator of a left ideal A in a ring R FF = F(R) = {A | A is left-max in R} ∼ class(M) class(M) = {RN | RN = RM} {Ki | i ∈ I} (SDR) system of distinct representatives of the isomorphism classes of ideal-simple left R-modules C(R) C = C(R) = {class(Ki) | i ∈ I}, where {Ki | i ∈ I} is an SDR wl(R) left width of a ring R J(R) Jacobson radical of a ring R D(R) set of idempotent elements in a ring R ureg(R) set of unit-regular elements in a ring R U(R) set of unit elements in a ring R Zr(R) right singular ideal of a ring R {∗} the class of ∗ rings; for example {local} denotes the class of local rings L denotes an arbitrary left idealtor SR1 stable range 1 left UG left uniquely generated IC internal cancellation DF directly ﬁnite ¯ Kp Kaplansky’s subring; Kp = {(n, λ) ∈ Z × Zp[x] | λ(0) =n ¯} ¯ where p ∈ Z is a prime number and k = k + pZ ∈ Zp for all k ∈ Z SS(R) = {L | L is a left ideal of R} that aﬀords the SR1 rings

viii PP(R) = {Ra | a ∈ R} that affords the SR1 rings AA(R) = {l(a) | a ∈ R} that affords the left UG rings DD(R) = {Re | e2 = e ∈ R} that affords the IC rings JJ (R) = {L ⊆ J(R) | L is a left ideal of R} that affords the DF rings J1 J1(R) = {J(R)} that affords the DF rings J2 J2(R) = {L | L is a nil left ideal of R} that affords the DF rings J3 J3(R) = {L | L is a nilpotent left ideal of R} that affords the DF rings J4 J4(R) = {L | L is a left quasi-regular ideal of R} that affords the DF rings max XX (R) = {L | L ⊆ RR} that affords the class of left-max stable rings CC(R) = {R} that affords {rings} SL(R) set of left L-stable elements in a ring R SS (R) set of SR1 elements in a ring R SA(R) set of left UG elements in a ring R SD(R) set of IC elements in a ring R SJ (R) set of DF elements in a ring R M ≥c LM is a cover of L c M LM does not cover L M ≡ L M and L are equivalent; they afford the same class of rings M 6≡ L M and L are not equivalent L¯ closure of the left idealtor L

ix Part I

L-stability in Rings

1 Chapter 1

Introduction: Deﬁnitions, Examples and Fundamental Properties

In 1964, in his seminal work [9] on algebraic K-theory, Hyman Bass deﬁned a ring R to have stable range 1 (SR1) if every element a in R is SR1, that is a satisﬁes the following condition:1

Ra + L = R, a ∈ R and L is a left ideal of R, implies that a − u ∈ L for some unit u of R.

Here, we restrict the choice of the left ideal L to a prescribed set L(R) of left ideals of the ring R, and investigate the corresponding “L-stable” rings.

In this chapter, we lay the foundations of L-stability notion for rings. In the first section, we mention some advantages for introducing this new class of rings. After that, we introduce the notion of a left idealtor, we define the notion of left L-stability and give five main examples for left L-stable rings: the class of SR1 rings, the class of left UG rings, the class of IC rings, the class of directly finite rings, and the class of all rings. Next, we study two important properties of left idealtors that will be used frequently later. The last section discusses some basic useful results for L-stability that will be used repeatedly in the other chapters.

1.1 Importance of introducing the L-stability Notion

It is a natural question that one may ask; why do we introduce this general notion of L-stability for rings?

1We need not say “left” SR1 because Vaserstein proved that this condition is left-right symmetric for rings, see [108] and [109].

2 The most important reason for doing so is that L-stability generalizes four large important well known classes of rings: the class of SR1 rings, the class of left UG rings, the class of IC rings and the class of directly finite rings. We apply the results we get in general for L-stability (often motivated by results for SR1) to these four particular classes of rings. Moreover, some of the results for these rings have been discussed in the literature, but here they follow quickly and easily with simpler arguments. On the other hand, we get new important results for these classes because the general notion of L-stability makes it easier to capture new properties for them than using the original definitions. In particular, this gives a new perspective on the directly finite rings and the open questions related to them. We will also be able to answer some open questions that have been stated for these rings in the literature. For example, we answer Question 1, Question 2 and Question 4 stated in [95] for the class of left UG rings.

1.2 Deﬁnitions and Four Major Examples

We start by introducing a fundamental notion which plays a major rule in our investigation for L-stability:

Deﬁnition 1.2.1. A left idealtor is a map L that associates to every ring R a well-deﬁned, non-empty set L(R) of left ideals of R.

Examples of left idealtors include the following ﬁve key left idealtors which will be intensely used in what follows:

Deﬁnition 1.2.2 (Notation). S, P, D, A and J which are deﬁned, respectively, for each ring R by:2

1. S(R) = {L | L is a left ideal of R}.

2. P(R) = {Ra | a ∈ R}.

3. D(R) = {Re | e2 = e ∈ R}.3

4. A(R) = {l(a) | a ∈ R}.

5. J (R) = {L ⊆ J(R) | L is a left ideal of R}.

2We abuse the notation as in Calculus where one speaks of the function f(x) = x2 + 1. Moreover, in what follows, S, P, D, A and J will denote the left idealtors deﬁned here. 3An element a in a ring R is called an idempotent if a2 = a.

3 Lemma 1.2.3. Let L be a left idealtor, and let R be a ring. Then, the following two statements are equivalent for an element a ∈ R:

1. If Ra + L = R where L ∈ L(R), then a − u ∈ L for some unit u in R.

2. If Ra + L = R where L ∈ L(R), then the coset a + L contains a unit of R.

Definition 1.2.4. Let L be a left idealtor. An element a in a ring R is called left L-stable if a satisfies the conditions in Lemma 1.2.3, and a ring R is called left L-stable if every element a in R is left L-stable. Moreover, we say that a class C of rings is afforded by a left idealtor L if C is the class of all left L-stable rings. Finally, we say that a class C of rings is affordable if it is afforded by some left idealtor L.

Because we discuss many classes of rings, we will use the following notation: a class of rings with a name will be denoted by {name}; for example {commutative} is the class of commutative rings. Moreover, for a left idealtor L, every conclusion for left L-stable rings has an obvious “right” version. So, from now on we will conﬁne ourselves to left L-stable rings. In addition, we ﬁx the following notations that will be used repeatedly:

Notation 1.2.5. The sets U(R), D(R) and ureg(R) will denote, respectively, the set of all units in R, the set of all idempotent elements in R and the set of all unit-regular elements in R.

As mentioned earlier, a ring R is SR1 if and only if the following condition holds:

Ra + L = R, a ∈ R and L is a left ideal of R, implies that a − u ∈ L for some u ∈ U(R).4

It follows that the class of SR1 rings is affordable, and it is afforded by the left idealtor S defined by

S(R) = {L | L is a left ideal of R} for each ring R.

Hence, we have {SR1} = {left S-stable}.

The following simple fact will be used frequently below.

Lemma 1.2.6. Let M and L be two left idealtors. If M(R) ⊇ L(R) for each ring R, then we have {left M-stable} ⊆ {left L-stable}.

4SR1 rings have been intensely studied by many authors. See for example [4], [5], [31], [50], [73], [112] and [117].

4 Proof. Let R be left M-stable. Suppose that Ra + L = R with L ∈ L(R) and a ∈ R. Then, by assumption, L ∈ M(R) and hence a − u ∈ L for some unit u in R because R is left M-stable. Hence, a is left L-stable, and so R is left L-stable, as required.

It is well-known that the left idealtor S is not the only one aﬀording the SR1 rings as the following example veriﬁes:

Example 1.2.7. The SR1 rings are aﬀorded by the left idealtor P(R) = {Rb | b ∈ R}.

Proof. Since {left S-stable} = {SR1}, we show that {left P-stable} = {left S-stable}. Assume that R is left P-stable. To see that R is left S-stable, let Ra + L = R with a ∈ R and L ∈ S(R), say ra + b = 1 where r ∈ R and b ∈ L. Thus, Ra + Rb = R which implies that a − u ∈ Rb ⊆ L for some unit u in R because a is left P-stable. Hence, {left P-stable} ⊆ {left S-stable}. Now, because P(R) ⊆ S(R) for every ring R, we have {left S-stable} ⊆ {left P-stable} by Lemma 1.2.6, as required.

A ring R is called a local ring if R/J(R) is a division ring. An element a in a ring R is called regular if aba = a for some b ∈ R, and a ring R is called a regular5 ring if every element has the property. A regular reduced ring is called strongly regular. Finally, a ring R is called unit-regular if every element a in R is unit-regular, that is, a = aua for some unit u in R [42].6

In addition to the class of SR1 rings, we have following two important, well-known, aﬀordable classes of rings:

The Class of Left Uniquely Generated Rings

An element a in a ring R is called left uniquely generated (left UG) if Ra = Rb, b ∈ R, implies b = ua for some u ∈ U(R), and R is called a left UG ring if every element in R is left UG. These rings were introduced in 1949 by Kaplansky [61] in his classic paper on elementary divisors. He showed that domains and local rings are left UG rings, and gave an example of a commutative ring that was not UG. It is still an open question whether or not the UG condition is a left-right symmetric property for rings. Moreover, these rings are also discussed in [95], and a detailed study of commutative UG rings can be found in [1]. In 1995, Canfell [23] showed that the left UG rings are aﬀorded by the left idealtor A deﬁned by

5Also called von Neumann regular ring. These rings have been studied intensely, see for example [49] and [47]. 6See [34], [36] and [118] for a module version of the regularity and the unit-regularity conditions.

5 A(R) = {l(b) | b ∈ R} for each ring R.

The Class of Internal Cancellation Rings

A module M is said to have internal cancellation if, whenever we have M = K ⊕ N = K0 ⊕ N 0 where K =∼ K0 then necessarily N =∼ N 0.7 In 1976, Ehrlich [43] showed that for a ring R, the left module RR has internal cancellation if and only if every regular element in R is unit-regular. Thus, “internal cancellation” is a left-right symmetric condition for rings. In 2005, Khurana and Lam [64] called these rings IC rings and gave a good survey of them.8 Earlier, in 2002, H. Chen [29], showed that these rings are aﬀorded by the left idealtor D deﬁned by

D(R) = {Re | e2 = e ∈ R} for each ring R.

As another example, the class of all rings is aﬀorded by the left idealtor C(R) = {R} as one can easily check.

The Class of Directly Finite Rings is Aﬀordable

A ring R is called a directly ﬁnite ring (a DF ring) if ab = 1 in R implies that ba = 1. Examples include the commutative rings.9 Moreover, a submodule K of a left R-module M is said to be small in M, denoted by K ⊆sm M, if K + X = M where X is a submodule of M, implies that X = M. Moreover, it is well known that J(R) is small in both RR and RR, and it contains every small one-sided ideal of R.

We now show that the class of DF rings is aﬀorded by some left idealtors, which gives a new perspective on these rings, as follows:

Theorem 1.2.8. The class {DF} is aﬀorded by the left idealtors J and T deﬁned respectively, for each ring R, by:

J (R) = {L ⊆ J(R) | L is a left ideal of R} and T (R) = {0}.

Proof. To prove the result, we verify the following inclusions:

7See for example [11], [13] and [52] 8IC rings have also been called partially unit-regular rings in [53]. 9These rings also called Dedekind finite, or von Neumann finite. Moreover, it is clear, using the definition, that the “DF” condition is a left-right symmetric property for rings.

6 {DF} ⊆ {left J -stable} ⊆ {left T -stable} ⊆ {DF}.

Since we have J (R) ⊇ T (R) for each ring R, it follows that {left J -stable} ⊆ {left T -stable} by Lemma 1.2.6. So, it remains to prove the following two inclusions:

•{ DF} ⊆ {left J -stable}. Assume that R is a DF ring, and let Ra + L = R where a ∈ R and sm L ∈ J (R). Now, since L ⊆ J(R) ⊆ RR, we have Ra = R which implies that a ∈ U(R) because R is a DF ring. Hence, a − u ∈ L with u = a which implies that a is left J -stable. Therefore, R is a left J -stable ring, as required.

•{ left T -stable} ⊆ {DF}. Let ba = 1 in R, we need to show that a is a unit in R. Now, since ba = 1 we have Ra + 0 = R. As 0 ∈ T (R), it follows that a − u ∈ 0 for some u ∈ U(R) because R is left T -stable. Therefore, a = u is a unit in R which implies that R is a DF ring, as required.

Corollary 1.2.9. The class {DF} is also aﬀorded by the following left idealtors:

1. J1(R) = {J(R)}.

2. J2(R) = {L | L is a nil left ideal of R}.

3. J3(R) = {L | L is a nilpotent left ideal of R}.

4. J4(R) = {L | L is a left quasi-regular ideal of R}.

Proof. In each case, we have

J (R) ⊇ Ji(R) ⊇ T (R) for each ring R, where T (R) = {0} and J (R) = {L ⊆ J(R) | L is a left ideal of R}. Therefore, the result follows using Theorem 1.2.8 and Lemma 1.2.6.

Observe that each of the left idealtors mentioned above that aﬀord the class of DF rings contains a left ideal in J(R). So, the following question is of interest:

Question 1.2.10. Let L be a left idealtor that aﬀords the class of DF rings. Must L(R) contain a left ideal in J(R) for every ring R?

7 We make the following useful deﬁnitions:

Deﬁnition 1.2.11. For any ring R, we deﬁne SS (R) = {a ∈ R | a is SR1}. Similarly, we have

SA(R) = {a ∈ R | a is left UG}, SD(R) = {a ∈ R | a is IC} and SJ (R) = {a ∈ R | a is DF}. More generally, for any left idealtor L, we deﬁne SL(R) by

SL(R) = {a ∈ R | a is left L-stable}.

For an easy reference we have:

Example 1.2.12. Each of the following ﬁve classes of rings is aﬀorded by the corresponding left idealtor:10

1. The class of all rings is aﬀorded by C(R) = {R}.

2. The SR1 rings are aﬀorded by S(R) = {L | L is a left ideal of R} and P(R) = {Rb | b ∈ R}.

3. The Left UG rings are aﬀorded by A(R) = {l(b) | b ∈ R}.

4. The IC rings are aﬀorded by D(R) = {Re | e2 = e ∈ R}.

5. The DF rings are aﬀorded by the following set of left idealtors:

(a) J (R) = {L ⊆ J(R) | L is a left ideal of R}.

(b) J1(R) = {J(R)}.

(d) J3(R) = {L | L is a nilpotent left ideal of R}.

(e) J4(R) = {L | L is a left quasi-regular ideal of R}.

(g) T (R) = {0}.

Special Rings

We end this section by listing the following four important examples of rings that will be used repeatedly as counter examples in what follows:

10In what follows, we fix and relate each of these left idealtors with its corresponding affordable class of rings. For example, D will denote the left idealtor D(R) = {Re | e2 = e ∈ R} that affords the class of IC rings.

8 Example 1.2.13 (Kaplansky [61]). Let p ∈ Z be a prime number, write D = Zp[x] and denote ¯ k = k + pZ ∈ Zp for all k ∈ Z. Deﬁne, the subring Kp of Z × D as follows:

¯ Kp = {(n, λ) ∈ Z × D | λ(0) =n ¯}.

Then:

1. The ring K5 is IC and DF because it is commutative. Moreover, it is not left UG because

(0, x) and (0, 2¯x) generate the same ideal of K5, but they are not unit multiples [61]. Thus, it is neither SR1 nor unit-regular.11

2. If p ≥ 5, then the ring Kp is not left UG [95]. Hence, it is neither SR1 nor unit-regular. But it is IC and DF being commutative.

3. The two rings K2 and K3 are IC and DF being commutative. On the other hand, they are UG rings as discussed in [95].

A ring R is called left artinian if, whenever we have L1 ⊇ L2 ⊇ · · · where each Li is a left ideal of R, then Ln = Ln+1 for some n ∈ N. And a ring R is called semilocal if R/J(R) is a left artinian ring. Moreover, R is called semisimple if every left R-module M is semisimple, that is,

M = ⊕i∈I Mi where each Mi is a simple submodule of M. Finally, a ring R is called semiperfect if R/J(R) is semisimple and J(R) is lifting [88].12

Example 1.2.14. For any two distinct primes p, q ∈ Z, deﬁne the ring R as follows:

a R = Z(p,q) = { b ∈ Q | p - b, q - b}.

Then, R is left UG being a domain [61]. R is clearly semilocal because

∼ Z(p,q)/J(Z(p,q)) = Zp × Zq.

Moreover, R is SR1, left UG, IC and DF13. On the other hand, R is not a semiperfect ring because idempotents do not lift modulo J(R).

Example 1.2.15. Let R = M2(D) be the 2 × 2 matrices over a division ring D. Then, R is SR1 because D is SR1 [73], and hence left UG, IC and DF. Moreover, R is unit-regular [49].

11Commutative rings are IC by [64]. SR1 and unit-regular rings are left UG by Theorem 1.4.14. 12We say that an ideal A of a ring R is lifting if every idempotent in R/A can be lifted to R, that is, for any idempotent element r + A in R/A, there exists e ∈ D(R) such that e − r ∈ A. 13R/J(R) is SR1 using Corollary 2.3.2. Hence, R is SR1 using Corollary 2.1.8, and so R is left UG, IC and DF using Theorem 1.4.14.

9 R is called a semi-regular ring if R/J(R) is regular and J(R) is lifting.14 A ring R is called an exchange ring if for any element a in R, there exists an idempotent e2 = e ∈ Ra with 1 − e ∈ R(1 − a), equivalently,

R = A + L, A and L are left ideals of R, implies that e2 = e ∈ A exists with 1 − e ∈ L.

This condition is left-right symmetric for rings using [89] or [110]. It is also known that a ring R is exchange if and only if every left (right) ideal of R is lifting [86]. Finally, the next example will be used frequently in what follows:

Example 1.2.16. Consider the ring of integers Z. Then:

1. Z is neither regular nor unit-regular.

2. Z is not SR1.

3. Z is a left UG ring, and so A(Z) = {0, Z}.

4. Z is an IC ring, and thus D(Z) = {0, Z}.

5. Z is a DF ring, and hence J (Z) = {0}.

Proof. (1). If a = 2 ∈ Z, then there is no element b ∈ Z such that a = aba. Hence, 2 is not a regular element in Z, and so Z is not a regular ring. Hence, Z is not unit-regular. (2). It is clear that Z2 + Z5 = Z and U(Z) = {1, −1}, but there is no u ∈ U(Z) such that 2−u ∈ Z5. Hence, Z is not left P-stable, where the left idealtor P is deﬁned by P(R) = {Rb | b ∈ R}. Therefore, Z is not SR1 because the class of SR1 rings is aﬀorded by P using Example 1.2.7, as required.15

(3). This follows as Z is a domain [61]. (4) and (5). These follow as Z is commutative.

1.3 Two Properties of Left Idealtors

Deﬁnition 1.3.1. Let M and L be two left idealtors. We say that M covers L, denoted by M ≥c L, if the following condition holds for each ring R:

b ∈ L ∈ L(R) implies that b ∈ M ⊆ L for some M ∈ M(R). (1.1)

14For more details about these rings, see [84]. These rings have a generalization in [102]. 15 Later in Example 3.3.7, we show that SS (Z) = {0, 1, −1}.

10 Example 1.3.2. Consider the two left idealtors

S(R) = {L | L is a left ideal of R} and P(R) = {Rb | b ∈ R} that aﬀord the class of SR1 rings. Then, we have S ≥c P and P ≥c S as one can check.

We generalize Lemma 1.2.6 as follows:

Proposition 1.3.3. Let M and L be any two left idealtors. Then:

1. If M(R) ⊇ L(R) for each ring R, then M ≥c L.

2. If M ≥c L, then {left M-stable} ⊆ {left L-stable}.

Proof. (1). Assume that M(R) ⊇ L(R) for each ring R. If b ∈ L ∈ L(R), then b ∈ M ⊆ L where M = L ∈ M(R). This proves that M ≥c L. (2). Let R be a left M-stable ring. If Ra + L = R where a ∈ R and L ∈ L(R), then ra + b = 1 for some r ∈ R and b ∈ L. Because M ≥c L, we have b ∈ M ⊆ L for some M ∈ M(R). Hence, 1 = ra + b ∈ Ra + M, and so Ra + M = R. Now, as R is left M-stable, we have a − u ∈ M for some u ∈ U(R). Since M ⊆ L, it follows that a is left L-stable, and hence R is a left L-stable ring. Therefore, we have {left M-stable} ⊆ {left L-stable}, as required.

As we have seen in Example 1.2.12 different left idealtors can afford the same class of rings. This motivates us to suggest the following definition:

Deﬁnition 1.3.4. Let M and N be two left idealtors. We say that M and N are equivalent, denoted by M ≡ N , if they aﬀord the same class of rings, that is, {left M-stable} = {left N -stable}.

Using Example 1.2.12 we have:

Example 1.3.5. The left idealtors S and P are equivalent since they both aﬀord the class of SR1 rings. Moreover, the left idealtors J , J1, J2, J3, J4 and T are all equivalent as they aﬀord the class of DF rings.

The next useful results give two equivalent modiﬁcations of a given idealtor:

Proposition 1.3.6. If L is any left idealtor, deﬁne the new left idealtor LJ by

LJ (R) = {L + C | L ∈ L(R),C ⊆ J(R)} for each ring R.

Then, we have L ≡ LJ .

11 Proof. Since L(R) ⊆ LJ (R) for each ring R, we have {left LJ -stable} ⊆ {left L-stable} using Proposition 1.3.3. Now, assume that R is a left L-stable ring. Let Ra + X = R with a ∈ R and X ∈ LJ (R), and write X = L + C with L ∈ L(R) and C ⊆ J(R). Then, Ra + (L + C) = R, and so sm Ra + L = R as C ⊆ J(R) ⊆ RR. Hence, a − u ∈ L for some u ∈ U(R) by hypothesis. As L ⊆ X, we have a is a left LJ -stable element, and so we have R is a left LJ -stable ring. Thus, it follows that {left L-stable} ⊆ {left LJ -stable}, and so L ≡ LJ , as required.

Proposition 1.3.7. Let L be any left idealtor. Deﬁne the left idealtor L1 by

L1(R) = L(R) ∪ {R} for each ring R.

Then, we have L ≡ L1.

Proof. We have {left L1-stable} ⊆ {left L-stable} using Proposition 1.3.3 because L(R) ⊆ L1(R) for each ring R. Next, assume that R is a left L-stable ring, and let Ra + X = R with a ∈ R and X ∈ L1(R). If X ∈ L(R), then a − u ∈ X for some u ∈ U(R) because R is left L-stable; if X = R, then a − 1 ∈ X. Hence, R is left L1-stable, and so {left L-stable} ⊆ {left L1-stable}. Therefore, we have L ≡ L1, as required.

In the next result, we get a suﬃcient condition for two left idealtors to be equivalent using the notion of covers as follows:

Proposition 1.3.8. If M ≥c L and L ≥c M, then M ≡ L.

Proof. If M ≥c L, then {left M-stable} ⊆ {left L-stable} follows by Proposition 1.3.3. Also, since L ≥c M, then {left L-stable} ⊆ {left M-stable} again using Proposition 1.3.3. Hence, we have {left M-stable} = {left L-stable} which implies that M ≡ L, as required.

Notice that:

Remark 1.3.9. The converse of Proposition 1.3.8 need not be true in general. Indeed, as mentioned in (2) of Example 1.3.5, the following three left idealtors are equivalent

J (R) = {L ⊆ J(R) | L is a left ideal }, J1(R) = {J(R)} and T (R) = {0}.

c c On the other hand, we have J ≥ T , but T J and neither J1 nor T covers each other for any ring R with J(R) 6= 0.

So, the following question is of interest:

Question 1.3.10. Can Deﬁnition 1.3.1 be restated so that the converse holds in Proposition 1.3.8?

12 1.4 Basic Features and Essential Results for L-stability

In this main section, we investigate and study the basic properties and the main features for the notion of L-stability, and prove some results that will be useful to use in later chapters.

In Section 1.2, we have mentioned four main aﬀordable classes of rings: the class of SR1 rings, the class of left UG rings, the class of IC rings and the class of DF rings. The next useful result investigates the relation between these classes of rings as follows:

Proposition 1.4.1. The following strict inclusions hold:

{SR1} ⊂ {left UG} ⊂ {IC} ⊂ {DF}.

Proof. The classes of SR1 rings, left UG rings, IC rings and DF rings are aﬀorded, respectively, by the left idealtors

P(R) = {Rb | b ∈ R}, A(R) = {l(b) | b ∈ R}, D(R) = {Re | e2 = e ∈ R} and T (R) = {0} using Example 1.2.12. Moreover, it is clear that P(R) ⊇ A(R) ⊇ D(R) ⊇ T (R) for each ring R. Thus, we have

{left P-stable} ⊆ {left A-stable} ⊆ {left D-stable} ⊆ {left T -stable} using Proposition 1.3.3, that is, {SR1} ⊆ {left UG} ⊆ {IC} ⊆ {DF}. On the other hand, these inclusions are strict. For the ﬁrst inclusion, we have the ring of integers Z is a left UG ring that is not SR1 using Example 1.2.16. For the second inclusion, Kaplansky’s ring K5 in Example 1.2.13 is an IC ring that is not left UG. Finally, the ring R in Goodearl’s Example 5.10 [49] is a regular DF ring that is not unit-regular, and hence it is not IC by Ehrlich’s Theorem [43] which shows that the last inclusion is also strict. Therefore, we have {SR1} ⊂ {left UG} ⊂ {IC} ⊂ {DF}, as desired.

A ring R is said to be abelian if idempotents in R are central.

Proposition 1.4.2. For the class of commutative rings we have:

1. {commutative} ⊂ {IC} ⊂ {DF}.

2. Neither SR1 rings nor commutative rings implies the other.

3. Neither left (right) UG rings nor commutative rings implies the other.

13 Proof. (1). Abelian rings are IC using Example 2.1 in [64]. Hence, the ﬁrst inclusion follows because commutative rings are clearly abelian, and it is strict as the division rings are IC but not commutative. The second strict inclusion follows using Proposition 1.4.1.

(2). The ring of integers Z is a commutative ring that is not SR1 using Example 1.2.16. On the other hand, any division ring is SR1 but not commutative.

(3). Kaplansky’s ring K5 in Example 1.2.13 is a commutative ring that is neither left nor right UG. Moreover, R in Example 1.2.15 is a left and right UG ring that is not commutative.

1.4.1 A Useful Tool to Check Ring Aﬀordability

As we have seen earlier that the class of SR1 rings is aﬀordable. This class of rings also plays an important role among the aﬀordable classes of rings as we can see in the following key result that will be used repeatedly in what follows:

Theorem 1.4.3 (Aﬀordability Theorem). The following two statements hold for the class of SR1 rings:

1. The SR1 rings are left L-stable for any left idealtor L, but not conversely.

2. If C is an aﬀordable class of rings, then {SR1} ⊆ C.

Proof. (1). Consider the left idealtor S(R) = {L | L is a left ideal of R} that aﬀords the class of SR1 rings. Then, for any left idealtor L clearly we have S ≥c L. Thus, using Proposition 1.3.3, it follows that {SR1} = {left S-stable} ⊆ {left L-stable}. (1.2)

For the converse, consider the ring of integers Z in Example 1.2.16; it is left A-stable for the particular left idealtor A(R) = {l(a) | a ∈ Z} being a left UG ring, but it is not SR1. (2). Suppose that C is aﬀorded by a left idealtor L. It follows that

{SR1} = {left S-stable} ⊆ {left L-stable} = C, as required.

The following question is of interest:

Question 1.4.4. If C is a class of rings and {SR1} ⊆ C, must C be aﬀordable?

14 One virtue of the Aﬀordability Theorem 1.4.3 is that it makes it easier to identify when a class C of rings is not aﬀordable:

To show C is not aﬀordable, ﬁnd a SR1 ring R that is not in the class C. (1.3)

A ring R is called prime (semiprime) if, for ideals A and B of R, AB = 0 implies A = 0 or B = 0 (Ak = 0, k ≥ 1 implies A = 0). In addition, a ring R is said to be left P-injective if every

R-linear map Ra → R, a ∈ R, extends to RR→RR, and it is called left mininjective if every

R-linear map L →RR, L is a simple left ideal of R, extends to RR→RR (see [97] and [98]). Finally, a ring R is called left Kasch if every simple left R-module embeds in RR, equivalently, if r(L) 6= 0 for every maximal left ideal L of R [63].16

In the following example, we list some classes of rings that are not aﬀordable:

Example 1.4.5. Each of the following classes of rings is not aﬀordable:

1. {commutative}, {semilocal}.

2. Any class C of rings in which J(R) is lifting for each ring R in C.

3. {exchange}, {semiperfect}, {semi-regular}, {local}, {unit-regular}, {regular}.

4. {prime}, {semiprime}, {left mininjective}, {left P -injective}, {left Kasch}.

Proof. In each case we employ (1.3):

(1). The ring M2(R) is SR1 using Example 1.2.15. But it is not commutative, and it is also not semilocal using Bass’s theorem [8]. a (2). The ring R = Z(2,3) = { b ∈ Q | 2 - b, 3 - b} in Example 1.2.14 is SR1, but J(R) is not lifting. (3). Each of these classes of rings is exchange [86], and hence J(R) is lifting. Now, the ring R in (2) is SR1, but J(R) is not lifting. (4). Consider the upper triangular matrix ring R over a nonzero division ring D   DD R =  . 0 D

Then, R is SR1, but enjoys none of the properties in (4).17

16See [99] for more details about the left mininjective rings, the left P -injective rings and the left Kasch rings. 17R is SR1 follows using Corollary 2.6.21 that will discussed later in Section 2.6.

15 Corollary 1.4.6. If C ⊆ D are classes of rings and D is aﬀordable, then C need not be aﬀordable in general.

Proof. The class {DF} is aﬀordable using Theorem 1.2.8. On the other hand, the subclass {commutative} is not aﬀordable by (1) of Example 1.4.5.

1.4.2 DF Rings Revisited

As mentioned in Proposition 1.4.1, the SR1 rings, the left UG rings and the IC rings are all DF. So, we ask:

Question 1.4.7. Can this phenomenon be generalized? That is, if L is any left idealtor and R is a left L-stable ring, must R be a DF ring?

One partial answer to Question 1.4.7 appears in the following result:

Proposition 1.4.8. Suppose that L is a left idealtor where, for each ring R, L(R) contains a left ideal C ⊆ J(R). Then, {left L-stable} ⊂ {DF}.

Proof. Assume that R is left L-stable. By assumption, let C ∈ L(R) where C ⊆ J(R). To prove that R is DF, we let ba = 1 in R. Then, certainly we have Ra + C = R because Ra = R. Now, as R is left L-stable, there exists u ∈ U(R) such that a − u ∈ C. So, a = u + c where c ∈ C. Since c ∈ J(R), we have a is a unit too, and so ab = 1. Hence, {left L-stable} ⊆ {DF}. On the other hand, this inclusion is strict because the left idealtor S(R) = {L | L is a left ideal of R} that aﬀords the SR1 rings satisﬁes the assumption above, but the ring of integers Z is a DF ring that is not left S-stable because it is not SR1 ring using (2) of Example 1.2.16. Therefore, {left L-stable} ⊂ {DF}, as desired.

Next, we weaken the deﬁnition of left L-stability as follows:

Deﬁnition 1.4.9. Let L be any left idealtor. An element a ∈ R is said to be left L-gen-stable provided that

Ra + L = R with L ∈ L(R), implies a − x ∈ L for some left invertible element x in R.

And a ring R is called left L-gen-stable if every element a in R is left L-gen-stable.

Motivated by Vaserstein’s proof [109] that every SR1 ring is DF, we introduce the following new class of rings:

16 Deﬁnition 1.4.10. Let L be any left idealtor. An element a in a ring R is called left L-Vaserstein if a satisﬁes the following condition:

f 2 = f ∈ r(a) and 1 − f ∈ Ra implies Rf ∈ L(R).

Thus, a ring R is said to be left L-Vaserstein if every element a in R is left L-Vaserstein.

We are now ready to prove the main result in this part:

Theorem 1.4.11. Let L be a left idealtor. Then, the following statements are true for any left L-Vaserstein ring R:

1. If a ∈ R is left L-gen-stable and ab = 1 with b ∈ R, then ba = 1.

2. Every left L-gen-stable ring is DF.

3. R is left L-gen-stable if and only if R is left L-stable.

4. If R is left L-stable, then R is DF.

Proof. (1). Assume that a ∈ R is left L-gen-stable, and let ab = 1 with b ∈ R. If we write f = 1 − ba, then f 2 = f, 1 − f = ba ∈ Ra and af = a − aba = 0. As a is left L-Vaserstein, we have Rf ∈ L(R). Since ba + f = 1, then Ra + Rf = R. But, a is a left L-gen-stable element, so let

a − x ∈ Rf where x is a left invertible element in R.

Now, observe that fb = b − bab = 0, so (a − x)b ∈ Rfb = 0, that is, xb = ab = 1. Hence, x is right invertible too, and so x is a unit in R. But then b is also a unit because xb = 1. Therefore, a is a unit because ab = 1. It follows that ba = 1, as required. In fact, we have a = b−1 = x. (2), (3) and (4). These follow using (1) above and the deﬁnitions.

1.4.3 Special Left L-stable Elements

We ﬁrst observe that units are left L-stable for any left idealtor L. A condition c on an element a in a ring R is said to be a translation invariant if, whenever a satisﬁes c, then so do ua and au for every u ∈ U(R). For L-stability we have:

Lemma 1.4.12 (Translation Lemma). Let L be any left idealtor. Then, the following two statements are true for an element a in a ring R:18

18Note that this result will be generalized later in Section 3.3; where we prove that the product of any two left L-stable elements in a ring R is again left L-stable provided that L is a normal left idealtor.

17 1. If a ∈ SL(R), then so is ua for any u ∈ U(R).

2. If L is normal and a ∈ SL(R), then so is au for any u ∈ U(R).

Proof. (1). Assume that a is left L-stable and u is a unit in R. Let R(ua) + L = R with L ∈ L(R). It follows that Ra + L = R, and hence a − v ∈ L for some v ∈ U(R). Thus, ua − uv ∈ L where uv is a unit in R. Therefore, ua is left L-stable, as required. (2). Assume that L is normal. If a is left L-stable and u ∈ R is a unit, we need to show that au is also left L-stable. Now, if R(au) + L = R with L ∈ L(R), then Ra + Lu−1 = R. But, we have Lu−1 ∈ L(R) because L is normal, and so a − v ∈ Lu−1 for some unit v in R because a is left L-stable. Thus, au − vu ∈ L which implies that au is left L-stable, proving (2).

Recall that a submodule N of a left R-module M is said to be essential, denoted by N ⊆ess M, if N ∩X 6= 0 for every nonzero submodule X of M. In addition, the right singular ideal of a ring ess R is deﬁned by Zr(R) = {z ∈ R | r(z) ⊆ RR}. In 2005, Khurana and Lam [64] proved that every unit-regular element in a ring R is SR1. Part (2) of the next fundamental result is a far-reaching extension of this fact with a much simpler proof. This result also shows that other element-wise properties are left L-stable for any left idealtor L.

Theorem 1.4.13. Let L be any left idealtor. Then, the following statements are true for an element a in a ring R:

1. If a ∈ D(R), then a is left L-stable.

2. If a ∈ ureg(R), then a is left L-stable.

3. If a ∈ J(R), then a is left L-stable.

4. If a ∈ Zr(R), then a is left L-stable provided that r(L) 6= 0 for all L ∈ L(R) with L 6= R.

Proof. (1). Assume that a2 = a is an idempotent element in R, and let Ra+L = R with L ∈ L(R). Write ra + x = 1 with r ∈ R and x ∈ L, and deﬁne u = 1 − (1 − a)ra. It follows that u is a unit in R and

a − u = a − 1 + (1 − a)ra = (1 − a)(−1 + ra) = (1 − a)(−x) ∈ L.

Therefore, a is left L-stable in R, as required.

18 (2). Let a be a unit-regular element in R, say a = ava where v is a unit in R. But, if we deﬁne e := va, then we have e ∈ D(R) and a = v−1e. Hence, a is left L-stable using (1) above and the Translation Lemma 1.4.12, proving (2). sm (3). Assume that Ra + L = R where L ∈ L(R) and a ∈ J(R). Since Ra ⊆ J(R) ⊆ RR, it follows that L = R and hence a − u ∈ L for any u ∈ U(R) which implies that a is left L-stable in R, as desired.

(4). Suppose Rz +L = R where z ∈ Zr(R) and L ∈ L(R). Then, 0 = r(R) = r(z)∩r(L). Hence, r(L) = 0 because z ∈ Zr(R). By hypothesis we have L = R, and so z − u ∈ L for any u ∈ U(R) which ends the proof of (4).

Finally, we have the following result which compares the ﬁve main classes of rings discussed so far:

Theorem 1.4.14. The following strict inclusions hold:

{unit-regular} ⊂ {SR1} ⊂ {left UG} ⊂ {IC} ⊂ {DF}.

Proof. The ﬁrst inclusion follows using Theorem 1.4.13, and it is strict as the upper triangular matrix ring over a division ring is SR1 but not unit-regular.19 The other strict inclusions follow by Proposition 1.4.1.

So, the following question is of interest:

Question 1.4.15. When do the classes of rings mentioned in Theorem 1.4.14 coincide?

1.4.4 A New Look at Three Old Well-Known Results

While the following three facts are not new, we discuss them here for completeness and use the results that we have shown earlier to give new shorter proofs for the ﬁrst two, and an element-wise version of the third:

Theorem 1.4.16. The following statements are equivalent for a semiperfect ring R:

1. R is SR1.

19Any upper triangular matrix ring R over a division ring D is SR1 using Corollary 2.6.21 that will discussed later in Section 2.6.

19 2. R is left UG.

3. R is IC.

Proof. (1) ⇒ (2) ⇒ (3). These implications follow using Proposition 1.4.1. (3) ⇒ (1). The IC rings are afforded by the left idealtor D(R) = {Re | e2 = e ∈ R}. Hence, these rings are also afforded by the left idealtor DJ (R) = {Re + C | e2 = e ∈ R,C ⊆ J(R)} using Proposition 1.3.6. But, DJ (R) is the set of all left ideals of any semiperfect ring R by Bass’ Lemma in [8]. Therefore, DJ affords the class of SR1 rings, proving (1).

Fuchs [48], Kaplansky [62] and Henriksen [55] showed independently that every regular SR1 ring is unit-regular. The same result also proved in [64]. In addition, Marks [79] also proved that a regular ring R is unit-regular if and only if R is left UG. The following result gathers these facts with a short quick proof:

Theorem 1.4.17. The following statements are equivalent for a regular ring R:

1. R is unit-regular.

2. R is SR1.

3. R is left UG.

4. R is IC.

Proof. (1) ⇒ (2) ⇒ (3) ⇒ (4). These implications follow by Theorem 1.4.14. (4) ⇒ (1). By deﬁnition, we have a ring R is IC provided that regular elements in R are unit-regular [43]. Thus, (1) follows as R is regular by assumption.

Using Theorem 1.4.16 and Theorem 1.4.17 we immediately get:

Corollary 1.4.18. Let R be a regular (semiperfect) ring. Then, R is left UG if and only if R is right UG.

One may expect that the DF property can be added to the statements in Theorem 1.4.17. But, unfortunately this is not the case because there are regular DF rings which are not unit-regular using Example 5.10 and Example 5.12 in [49].

Notice that the class of exchange rings generalizes the classes of regular rings and semiperfect rings. On the other hand, it is also well-known that the “semiperfect” and the “regular” requirements in Theorem 1.4.17 and Theorem 1.4.16 can be weakened to “exchange” (see for example

20 Theorem 6.5 in [64]).20 In the next result, we give a version of this fact for elements. To this end, we need the following deﬁnition:

Deﬁnition 1.4.19. Call an element a in a ring R left exchange if a satisﬁes the following condition:

Ra + L = R, L is a left ideal of R, implies e2 = e ∈ Ra exists with 1 − e ∈ L.

So, a ring R is exchange if and only if every element a in R is left exchange.

With this in mind we can give now an element-wise version of Theorem 6.5 in [64] as follows:

Proposition 1.4.20. If a is a left exchange element in a ring R, then the following statements are equivalent:

1. a is left SR1.

2. a is left UG.

3. a is left IC.

Proof. (1) ⇒ (2) ⇒ (3). These implications are true using the element-wise version of Theorem 1.4.14. (3) ⇒ (1). Assume that a is both left IC and left exchange, and let Ra + L = R where L is a left ideal of R. Since a is left exchange, we choose e2 = e in R with e ∈ Ra and 1 − e ∈ L. Now, as R = Re + R(1 − e) and Re ⊆ Ra, it follows that Ra + R(1 − e) = R. Hence, a − u ∈ R(1 − e) ⊆ L for some u ∈ U(R) because a is left IC by assumption, and so a is left SR1, proving (1).

Hence, we have:

Theorem 1.4.21. If R is an exchange ring, then:

1. The classes of SR1 rings, left UG rings and IC rings coincide.

2. R is left UG if and only if R is right UG.

Following Nicholson [86], R is said to be a potent ring if idempotents lift modulo J(R) and any left ideal of R that is not contained in J(R) contains a nonzero idempotent. So, one may expect that Theorem 1.4.21 holds for the class of potent rings which generalizes the class of exchange rings. Unfortunately, this is not the case as the following example veriﬁes:

20This result is also discussed in [21], [26], [27], [28] and [117].

21 Example 1.4.22. Consider the ring S = Q × Q × Q × · · · , and let R be the subring of S consisting of sequences of the form (x1, x2, . . . , xn, m, m, . . .) where n ≥ 1, m ∈ Z and xi ∈ Q. Then, R is a non-exchange potent ring [86]. Moreover, R is DF and IC, but it is not SR1 [64].

Proof. R is IC and DF using Proposition 1.4.2 because it is commutative. In addition, R is not

SR1 because R has an epimorphic image that is isomorphic to Z, where the latter ring is not SR1 using Example 1.2.16.21

1.4.5 An Extension of a Theorem of Bass

Theorem 1.4.13 asserts that unit-regular rings are SR1. On the other hand, matrix rings over division rings are unit-regular by Corollary 4.7 in [49]. It follows that semisimple rings are unit- regular, and so SR1 rings.22 With this in mind, we deﬁne a new class of rings as follows:

Deﬁnition 1.4.23. Call a ring R casilocal if R/J(R) is unit-regular.

Hence, we have:

Example 1.4.24. Every semilocal ring is casilocal. The converse fails.

Proof. By deﬁnition, R/J(R) is semisimple for any semilocal ring R. Thus, R/J(R) is unit-regular by the above discussion, and so R is casilocal. The converse fails because there are unit-regular rings which are not semisimple as Example 5.15 in [49] veriﬁes.

A noted theorem of Bass [9] asserts that all semilocal rings are SR1. Here is an extension:

Theorem 1.4.25 (Extended Bass Theorem). Every casilocal ring is SR1.

Proof. If R is casilocal, then R/J(R) is unit-regular. Hence, R/J(R) is SR1 by Theorem 1.4.13, and so R is SR1, as required.23

The following question is of interest:

Question 1.4.26. Does there exist an SR1 ring that is not casilocal? By Theorem 1.4.3 the answer is “no” if and only if {casilocal} is not aﬀordable.

21Notice that isomorphisms do not preserve L-stability in general as we will see Section 3.4. However, they do preserve the SR1 condition by Proposition 3.4.8 which will be proved later. 22Semisimple rings are unit-regular follows using the fact that the direct product of any unit-regular rings is again unit-regular, and the Wedderburn-Artin Theorem which asserts that a ring R is semisimple if and only if ∼ R = Mn1 (D1) × Mn2 (D2) × · · · × Mnk (Dk), where each Di is a division ring [87]. 23The fact that “a ring R is SR1 if and only if R/J(R) is SR1” and a generalization of this will be discussed later in Section 2.1.

22 We noticed earlier that the ring of integers Z is left UG, IC and DF using Example 1.2.16. On the other hand, Z is not casilocal because it is not unit-regular and J(Z) = 0. Hence, using this, Theorem 1.4.25 and Theorem 1.4.14 we have:

Corollary 1.4.27. Every casilocal ring is left UG, IC and DF. The converse is false.

There is another class of rings that plays a role here. Following Camillo and Yu [20], a ring R is called semi-unit-regular (SUR) if R/J(R) is unit-regular and J(R) is lifting. The connection with the present work is that R is SUR if and only if R is casilocal and J(R) is lifting.

So, we get:

Example 1.4.28. Every SUR ring is casilocal; the converse fails. Hence, SUR rings are SR1, left UG, IC and DF. Moreover, {SUR} is not aﬀordable.

Proof. As discussed above, every SUR ring is casilocal. The converse is not true because the ring a ∼ Z(2,3) = { b ∈ Q | 2 - b, 3 - b} in Example 1.2.14 is casilocal since R/J(R) = Z2 × Z3 is unit-regular, but R is not SUR as J(R) is not lifting. In addition, SUR rings are SR1, left UG, IC and DF because casilocal rings enjoy these properties as discussed above. Finally, {SUR} is not aﬀordable using Theorem 1.4.3 because the ring Z(2,3) is SR1 but not SUR.

We end this part proving the following result:

Proposition 1.4.29. The following inclusions hold:

{SUR}, {semilocal} ⊂ {casilocal} ⊆ {SR1} ⊂ {left UG} ⊂ {IC} ⊂ {DF}.

Proof. The result follows using Theorem 1.4.14, Example 1.4.24, Theorem 1.4.25 and Example a 1.4.28. Finally, the ring Z(2,3) = { b ∈ Q | 2 - b, 3 - b} in Example 1.2.14 is semilocal but not SUR; while any unit-regular ring that is not semisimple is SUR but not semilocal (see Example 5.15 in [49]).

23 Chapter 2

Related Rings

In this chapter, we discuss how left L-stable rings behave with respect to the following standard constructions in ring theory: Factor rings, corner rings, direct products, subrings, ideal extensions, polynomial rings, formal power series rings, and matrix rings.

2.1 Factor Rings

In this section, we verify that L-stability does not pass to factor rings in general. However, under certain conditions we show that factors ring of a left L-stable ring R is indeed left L-stable.

Example 2.1.1. The left UG, IC and DF properties do not pass to factor rings in general.

1 Proof. Consider the free algebra R = Qhx, yi [64]. Then, R is a left UG ring being a domain, and so it is an IC ring and a DF ring using Proposition 1.4.1. But, the factor ring of R obtained by using the relation xy = 1 is not a DF ring, and so it is neither IC nor left UG using again Proposition 1.4.1.

It follows that:

Proposition 2.1.2. L-stability does not pass to factor rings in general.

However, this is not the case for the SR1 condition as we can see in the following well-known fact [73]:2

1 For more information about the free algebra Qhx, yi and a generalization of this, we refer the reader to Example 1.2 in [71]. 2This fact stated without proof in Proposition 1.5 [73] for the general case “stable range n”. Here, we state and prove it for completeness for the particular case “stable range 1”.

24 Proposition 2.1.3. Every homomorphic image of any SR1 ring R is again SR1.

Proof. For simplicity, we prove the result for factor rings. Let R be SR1, and let R¯ = R/X be its factor ring where X is an ideal of R. Assume that, R¯a¯ + R¯¯b = R¯ witha, ¯ ¯b ∈ R¯. Then,r ¯a¯ +c ¯ = 1¯ wherer ¯ ∈ R¯ andc ¯ ∈ R¯¯b. Hence, (ra + c) + X = 1 + X, and then ra + c − 1 = x ∈ X. So, ra + (c − x) = 1, which implies Ra + R(c − x) = R. Now, as R is SR1, we have a − u ∈ R(c − x) for some u ∈ U(R). That is, a + t(c − x) = u for some t ∈ R, and so a + tc − u = tx ∈ X. Thus, a¯ + t¯c¯− u¯ = 0,¯ which implies thata ¯ − u¯ = −t¯c¯ ∈ R¯¯b whereu ¯ ∈ U(R¯). Therefore,a ¯ has stable range 1 in R¯, and hence R¯ is SR1, as required.

In addition to Proposition 2.1.3 it is also well-known that a ring R is SR1 if and only if R/J(R) is SR1 [73]. More generally, if ϕ : R → S is an onto ring morphism one may ask the following:

Question 2.1.4. Let L be a left idealtor. Assuming that R is a left L-stable ring, when must S be left L-stable (and conversely)?

The answer is closely related to the following conditions on the left idealtor L:

Deﬁnition 2.1.5. Let L be a left idealtor, and let ϕ : R → S be a ring morphism. Then:

1. ϕ is called L-ﬁt if L ∈ L(R) implies ϕ(L) ∈ L(S).

2. ϕ is called L-full if X ∈ L(S) implies X = ϕ(L) for some L ∈ L(R).

Because of the ubiquity of the coset map θ : R → R/A, A C R, we add:

3. An ideal A of a ring R is called L-ﬁt (L-full) if the coset map R → R/A is L-ﬁt (L-full).

With this in mind we prove the following fundamental result:

Theorem 2.1.6. Let L be a left idealtor, and let ϕ : R → S be an onto ring morphism such that ker(ϕ) ⊆ J(R). Then:

1. If ϕ is L-ﬁt and ϕ(a) is left L-stable in S with a ∈ R, then a is left L-stable in R.

2. If ϕ is L-full and a ∈ R is left L-stable in R, then ϕ(a) is left L-stable in S.

In particular: If S is left L-stable, then so is R provided that ϕ is L-ﬁt. If R is left L-stable, then so is S provided that ϕ is L-full.

25 Proof. The last two statements follow from (1) and (2). For convenience writer ¯ = ϕ(r) for any element r ∈ R. (1). Assume thata ¯ is left L-stable in S. To see that a is left L-stable in R, let Ra + L = R with L ∈ L(R), say ra + b = 1 where r ∈ R and b ∈ L. Then,r ¯a¯ + ¯b = 1¯ where ¯b ∈ L¯ ∈ L(S) as ϕ is L-ﬁt. So, we have Sa¯ + L¯ = S. Thus,a ¯ − u¯ ∈ L¯ whereu ¯ ∈ U(S) becausea ¯ is left L-stable in S by assumption. Let a − u − b ∈ ker(ϕ) where b ∈ L. Writing a − u − b = c we see that a − (u + c) = b ∈ L, and u + c ∈ U(R) because c ∈ ker(ϕ) ⊆ J(R). Hence, a is left L-stable in R, proving (1). (2). Assume now that a is left L-stable in R, and let Sa¯ +X = S with X ∈ L(S). As ϕ is L-full, write X = L¯ with L ∈ L(R). Then, Sa¯ + L¯ = S, say ra + b − 1 ∈ ker(ϕ) withr ¯ ∈ S and b ∈ L. It sm follows that Ra + L + ker(ϕ) = R. Hence, Ra + L = R because ker(ϕ) ⊆ J(R) ⊆ RR. As a is left L-stable in R, we have a − u ∈ L with u ∈ U(R). Therefore,a ¯ − u¯ ∈ L¯ andu ¯ is a unit in S, proving (2).

Despite Proposition 2.1.2, Theorem 2.1.6 asserts in particular that L-stability passes to factor rings under suitable assumptions as follows:

Corollary 2.1.7. Let A ⊆ J(R) be an ideal of a ring R. Then, the following statements are true for a left idealtor L:

1. If R/A is left L-stable, then so is R provided A is L-ﬁt.

2. If R is left L-stable, then so is R/A provided A is L-full.

As we have seen above, in Example 2.1.1, each of the ring properties left UG, IC and DF does not pass to factor rings in general. However, using Corollary 2.1.7, we show in the next result that these properties pass to special factor rings as follows:

Proposition 2.1.8. Let I ⊆ J(R) be an ideal of a ring R, and write R¯ = R/I. Then, the following statements are true:3

1. R is SR1 if and only if R¯ is SR1.

¯ 2. (a) If R is left UG, then so is R provided b ∈ R implies lR(b) = lR¯(¯c) for some c ∈ R.

3These facts have been discussed in the literature before; (1) in [73], (2) in [95], (3) in [64], and (4) is easily veriﬁed. However, here we give each of them a shorter proof than used before using the general notion of L-stability that we discuss and investigate.

26 ¯ (b) If R is left UG, then so is R provided c ∈ R implies lR¯(¯c) = lR(b) for some b ∈ R.

3. (a) If R¯ is IC, then so is R.

(b) If R is IC, then so is R¯ provided that idempotents lift modulo I.

4. R is DF if and only if R¯ is DF.

Proof. In each case, we verify that the assumptions of Corollary 2.1.7 hold: (1). For the SR1 rings, use the left idealtor S(R) = {L | L is a left ideal of R}. Thus, both assumptions are clear using the Correspondence Theorem.

(2). For the left UG rings, use A(R) = {lR(b) | b ∈ R}. So, the assumptions are, respectively, if b ∈ R, then lR(b) = lR¯(¯c) for some c ∈ R; and if c ∈ R, then lR¯(¯c) = lR(b) for some b ∈ R. (3). For the IC rings, use D(R) = {Re | e2 = e ∈ R}. Hence, the ﬁrst assumption is clear as Re = R¯e¯ ∈ L(R¯) for any e2 = e ∈ R. For the second, assume that R¯e¯ ∈ L(R¯) where e¯2 =e ¯ ∈ R¯. Since, idempotents lift modulo I, we may assume that e2 = e ∈ R. Thus, R¯e¯ = Re where Re ∈ L(R). (4). For the DF rings, use T (R) = {0}. So, both assumptions are clear in this case.

2.2 Corner Rings

In this section, we describe when L-stability of a ring R passes to its corner rings eRe for any e2 = e ∈ R as follows:

Theorem 2.2.1. Let L be any left idealtor, and let e ∈ D(R). If R is left L-stable, then so is eRe provided the following conditions hold:

1. If X ∈ L(eRe), then RX ∈ L(R).

2. One of the following two statements holds:

(a) Every left L-stable ring is DF.

(b) The map θ : R → eRe deﬁned by θ(r) = ere is a ring morphism.4

Proof. Let R be left L-stable, write S = eRe, and let Sa + X = S where a ∈ S and X ∈ L(S); we want a − w ∈ X for some unit w of S. Write sa + x = e, s ∈ S, x ∈ X. Then,

4Notice that the map θ is a ring morphism if and only if erse = erese for all r, s ∈ R.

27 (s + 1 − e)(a + 1 − e) + x = (sa + 1 − e) + x = 1.

Hence, R(a + 1 − e) + RX = R. Using (1), we have (a + 1 − e) − v := b ∈ RX for some v ∈ U(R) because R is left L-stable by assumption. Thus, we have:

(a + 1 − e − b)u = 1 where u := v−1 ∈ U(R). (2.1)

Multiply both sides by e to get (a − eb)ue = e. But, we have eb ∈ e(RX) = eR(eX) ⊆ X because X is a left ideal of S. In particular, b = be and it follows that

(a − eb)eue = e, eb ∈ X. (2.2)

Write w = a − eb, so w has a right inverse in S.

If (a) holds, it follows that w is a unit in S because S is DF whenever R is. But, as we have a − w = eb ∈ X, then a is left L-stable in S, as required.

Now assume (b). We show that eue ∈ U(S), and hence a − eb ∈ U(S) by (2.2). As in (2.1) we have u(a + 1 − e − b) = 1, whence eu(a − be) = e. Now condition (b) shows that eue(a − be) = e. This with (2.2) shows that eue is a unit in S, and we are done as before.

Corollary 2.2.2. Each of the ring properties SR1, left UG, IC and DF passes to corners.

Proof. The left UG condition passes to corners by Theorem 30 in [95]. Now, we use Theorem 2.2.1 to verify the result for the other properties. Notice that if R is SR1 or IC, then R is DF using Proposition 1.4.1. So, we only need to verify condition (1) in Theorem 2.2.1. Let S = eRe where e ∈ D(R). Then:

• For the SR1 rings, use S(R) = {L | L is a left ideal of R}. Hence, (1) is clear in this case.

• For the IC rings, use D(R) = {Re | e2 = e ∈ R}. Thus, (1) follows because RSf = Rf for any f ∈ D(S).

• For the DF rings, we may take T (R) = {0} and so (1) clearly holds.

28 2.3 Direct Products

First, we prove the following result about direct products:

Theorem 2.3.1. Let R = Πi∈I Ri denote a direct product of rings Ri with canonical projections

πk : R → Rk for each k ∈ I. Let L denote any left idealtor. Then:

1. If R is left L-stable, then so is each Ri provided Li ∈ L(Ri) for each i implies Πi∈I Li ∈ L(R).

2. If each Ri is left L-stable, then so is R provided L ∈ L(R) implies L = Πi∈I Li where each

Li ∈ L(Ri).

Proof. (1). Assume that R is left L-stable. Given i ∈ R, suppose that Riai + Li = Ri with

Li ∈ L(Ri) and ai ∈ Ri, say riai + xi = 1Ri where xi ∈ Li. Then, hriihaii + hxii = 1R with hxii ∈ Πi∈I Li ∈ L(R) by the proviso. Hence, haii − huii ∈ Πi∈I Li where huii is a unit in R because

R is left L-stable by assumption. Thus, ai − ui = si ∈ Li for each i, and each ui is a unit in Ri. This proves (1).

(2). Assume now that each Ri is left L-stable. Suppose Rhaii + L = R where L ∈ L(R). By the proviso, we have L = Πi∈I Li with Li ∈ L(Ri) for each i. Hence, hriihaii + hxii = h1Ri i with ri ∈ Ri and xi ∈ Li for each i. It follows that Riai + Li = Ri. Thus, by hypothesis, we have ai − ui ∈ Li for some unit ui in Ri. So, haii − huii ∈ Πi∈I Li = L where huii is a unit in R, proving (2).

Using Theorem 2.3.1 we immediately get the following well-known fact for the particular classes of SR1 rings, left UG rings, IC rings or DF rings:

Corollary 2.3.2. Let R = Πi∈I Ri denote a direct product of rings Ri. Then, R is SR1, left UG,

IC or DF if and only if the same is true for each Ri.

We now prove an interesting result for ﬁnite direct products. It is convenient to work internally:

R = S1 ⊕ · · · ⊕ Sn where Si C R for each i.

2 Hence, in this case Si = eiRei = Rei = eiR where for each i we have ei = ei is a central idempotent 5 in R, the set {e1, ··· , en} is orthogonal, and 1 = e1 + ... + en.

With this in mind, we prove the following:

5 The set {e1, ··· , en} of idempotents in a ring R is said to be orthogonal if eiej = 0 for any i 6= j.

29 Theorem 2.3.3. Let L denote any left idealtor, and assume that R = S1 ⊕ · · · ⊕ Sn where Si C R for each i. Then:

1. If R is left L-stable, then so is each Si provided L(Si) ⊆ L(R) for each i.

2. If each Si is left L-stable, then so is R provided {Si ∩ L | L ∈ L(R)} ⊆ L(Si) for each i.

2 Proof. Following the discussion above, write Si = eiRei where each ei = ei is a central idempotent in R, the set {e1, ··· , en} is orthogonal, and 1 = e1 + ··· + en.

(1). This follows from Theorem 2.2.1. Indeed, condition 2(b) is satisﬁed because the ei are central; and condition (1) holds because if X ∈ L(Si), then we have

RX = R(eiX) = SiX = X ∈ L(R) for each i by the assumption.

(2). Let Ra + L = R with a ∈ R and L ∈ L(R). Now, multiplying on the right by ei gives

Raei + Lei = Rei = Si. Observe that Lei = Si ∩ L ∈ L(Si) by the assumption, and we have

Raei = Reia = Sia. Hence, Sia + Lei = Si, and so Siaei + Lei = Si with Lei ∈ L(Si). Now, since each Si is left L-stable, there exists ui ∈ U(Si) such that aei − ui ∈ Lei for each i. Writing n u = Σi=1ui, it follows that u is a unit in R with inverse v = Σvi where uivi = ei = viui for each i.

Finally, we obtain a − u = Σ(aei − ui) ∈ ΣLei = ΣeiL ⊆ L, which ends the proof.

For the particular classes of SR1 rings, left UG rings, IC rings and DF rings we have:

Proposition 2.3.4. Let R = S1 ⊕ · · · ⊕ Sn where Si C R for each i. Then R enjoys each of the ring properties SR1, left UG, IC and DF if and only if the same is true of each Si.

Proof. If R is SR1, left UG, IC or DF, then so is each Si being a corner of R using Corollary 2.2.2. Now, using the left idealtors S(R) = {L | L is a left ideal of R} and T (R) = {0}; it is clear that assumption (2) of Theorem 2.3.3 holds for SR1 and DF. For the left UG condition, using the left idealtor A(R) = {lR(b) | b ∈ R}; assumption (2) of Theorem 2.3.3 holds because 2 Si ∩ lR(b) = lSi (b) ∈ A(Si). Finally for the IC property, using D(R) = {Re | e = e ∈ R}; assumption (2) of Theorem 2.3.3 also holds because for any idempotent element f 2 = f in R, we have Rei ∩ Rf = Reif = Sieif ∈ D(Si).

30 2.4 Subrings

In this section, we show that L-stability does not pass to subrings in general. However, in special cases we prove that L-stability passes to subrings assuming certain conditions. We also get some results for the particular classes of rings: SR1 rings, left UG rings, IC rings and DF rings.

2.4.1 Motivation

Recall that a subring S of a ring R is called unital if 1R ∈ S. First, we notice that:

Example 2.4.1. The ring R = Z × Z5[x] in Example 1.2.13 is a commutative ring which is also ¯ left and right UG, but its subring S = K5 = {(n, λ) ∈ Z × Z5[x] | λ(0) =n ¯} is neither left nor right ¯ UG, where k = k + 5Z ∈ Z5 for all k ∈ Z.

Example 2.4.2. There is a unit-regular ring R in Example 5.12 [49] which has a regular unital subring S that is not unit-regular. Note that R in this case is SR1, left UG, IC and DF using Theorem 1.4.14. On the other hand, its unital subring S is not IC by Ehrlich’s Theorem [43]6, and hence S is neither left UG nor SR1 using Proposition 1.4.1.

It follows that each of the ring properties SR1, left UG and IC does not pass to subrings in general. However, one can easily check that unital subrings of a DF ring R are again DF.

2.4.2 Ideal Extensions

If S is a unital subring of a ring R, then R is said to be an extension of S. In this part we examine a particular extension type and characterize when a stability condition passes from S to R, and conversely. The ring extensions we are interested in are described as follows:

Deﬁnition 2.4.3. A ring R is called an ideal extension of a unital subring S if R = S ⊕ A where 7 A C R and A ⊆ J(R).

For example the formal power series ring R = S[[x]] is an ideal extension of the ring S. Before proceeding, we need:

Lemma 2.4.4 (Shift Lemma). Let L be any left idealtor, and let R be a ring. If a ∈ SL(R), then so is a + c for any c ∈ J(R).

6Ehrlich’s Theorem asserts that a ring R is IC if every regular element in R is unit-regular. 7If the requirement that A ⊆ J(R) is dropped then R is called a Dorroh extension of S [40]. For example, the polynomial ring R = S[x] is a Dorroh extension of S because we have R = S ⊕ hxi.

31 Proof. Assume that a is left L-stable and c ∈ J(R). Let R(a + c) + L = R where L ∈ L(R). Then, r(a + c) + l = ra + rc + l = 1 with r ∈ R and l ∈ L. Hence, Ra + J(R) + L = R, which implies sm that Ra + L = R as J(R) ⊆ RR. Thus, a − u ∈ L, for some u ∈ U(R) as a is left L-stable by assumption. Therefore, we are done because (a + c) − (u + c) ∈ L with u + c ∈ U(R) since c ∈ J(R).

We prove the following main result:

Theorem 2.4.5. Let R = S ⊕ A be an ideal extension, and let L be any left idealtor. Deﬁne the map θ : R → S by θ(s + a) = s for all s ∈ S and a ∈ A. Then:

1. If R is left L-stable, then so is S provided θ is L-full.

2. If S is left L-stable, then so is R provided θ is L-ﬁt.

Proof. For clarity, we write L¯ = θ(L) for any left ideal L of R, andr ¯ = θ(r) for any r ∈ R. Note that θ is an onto ring morphism, ker(θ) = A, ands ¯ = s for any s ∈ S. Also, clearly we have U(S) ⊆ U(R); in fact U(R) = U(S) ⊕ A because A ⊆ J(R). (1). Let R be left L-stable, and let Sb + X = S where b ∈ S and X ∈ L(S), say 1 = sb + x with s ∈ S and x ∈ X. Now, as θ is L-full, X = L¯ where L ∈ L(R). Write, x = ¯l for some l ∈ L. Now, we have

1 − sb − l = x − l =x ¯ − ¯l =x ¯ − x = 0 because x ∈ S.

It follows that 1 − sb − l ∈ ker(θ) = A. Hence, Rb + L + A = R, from which Rb + L = R because sm ¯ A ⊆ J(R) ⊆ RR. By hypothesis, let b − u ∈ L where u ∈ U(R). As b = b, it follows that

b − u¯ = ¯b − u¯ = b − u ∈ L¯ = X.

Therefore, we are done sinceu ¯ ∈ U(S). (2). Assume now that S is left L-stable and let r ∈ R; we must show that r is left L-stable in R. Now, we have r = s + a with s ∈ S and a ∈ A, but since we have A ⊆ J(R), it suﬃces to show that s is left L-stable in R using the Shift Lemma 2.4.4. So, let Rs + L = R with L ∈ L(R), say ps+x = 1 with p ∈ R and x ∈ L. Hence, 1 = 1¯ =p ¯s¯+x ¯, which implies that S = Ss+L¯. Moreover, L¯ ∈ L(S) because θ is L-ﬁt. Thus, s − u ∈ L¯ for some u ∈ U(S) ⊆ U(R). Now, if s − u =x ¯ where x ∈ L, then s − u − x ∈ ker(θ) = A, say s − u − x = a ∈ A. Finally, s − (u + a) = x ∈ L, and we are done because u + a is a unit of R since A ⊆ J(R).

32 As we have noticed, in Example 2.4.2, that the notions of SR1, left UG and IC do not pass to subrings. However, we have the following positive case:

Proposition 2.4.6. Let R = S ⊕ A be an ideal extension. Then:

1. R is SR1 if and only if S is SR1.

2. If R is left UG, then so is S. The converse holds provided that for any b ∈ R, lR(b) = lR¯(¯c) for some c ∈ R, where R¯ = R/A andr ¯ = r + A for any r ∈ R.8

3. R is IC if and only if S is IC.9

4. R is DF if and only if S is DF.

Proof. As in Theorem 2.4.5, define the map θ : R → S by θ(s + a) = s for all s ∈ S and any a ∈ A. (1). Consider the left idealtor S(R) = {L | L is a left ideal of R}. Then, θ is S-fit because if L ∈ S(R), then θ(L) is a left ideal of S, and hence θ(L) ∈ S(S). In addition, θ is also S-full because if X ∈ S(S), then X = θ(X ⊕ A). Thus, we are done using Theorem 2.4.5. (2). Let R be a left UG ring. Then, S is left UG follows by applying (1) of Theorem 2.4.5 to the left idealtor A(R) = {l(b) | b ∈ R} because θ is A-full as if b ∈ S, then θ[lR(b)] = lS(b). Assume now that S is a left UG ring. Then, R/A is left UG because S =∼ R/A, and hence R is left UG by part (a) of (2) in Proposition 2.1.8 using the assumptions.10 (3). If R is IC, then so is S using (1) of Theorem 2.4.5 because if D(R) = {Re | e2 = e ∈ R}, then θ is D-full as θ(Re) = Se for any e2 = e ∈ S. Conversely, if S is IC, then so is R/A since S =∼ R/A, and thus R is IC by (3) of Proposition 2.1.8 because A ⊆ J(R) by assumption.11 (4). The result follows by applying Theorem 2.4.5 to the left idealtor T (R) = {0} because θ is clearly both T -full and T -fit.

2.5 Polynomial Rings and Rings of Formal Power Series

In this section, we discuss the passage of L-stability to formal power series rings and the polynomial rings.

8It has been shown in Proposition 10 [95] that if the Dorroh extension R = S ⊕ A is left UG, then so is S. Here, we prove this and its converse for the ideal extension with a simple argument using the notion of L-stability. 9It has also been shown in Proposition 5.1 [64] that if the Dorroh extension R = S ⊕ A is IC, then so is S. Here, we prove this and its converse for the ideal extension with a simple argument using the notion of L-stability. 10Notice that isomorphisms do not preserve L-stability for certain bizarre idealtors L as we will see in Section 3.4. However, they preserve the left UG condition using Proposition 3.4.8 that will be proved later. 11Isomorphisms also preserve the IC condition using Proposition 3.4.8.

33 We denote the ring of formal power series over the ring S by R = S[[x]]. With this in mind, we prove the following result, where (4) gives a partial answer to Question 2 in [95]:

Proposition 2.5.1. Let L be any left idealtor. For an ideal extension R = S ⊕ A, we deﬁne the map θ : R → S by θ(s + a) = s for all s ∈ S and any a ∈ A. Then:

1. If R is left L-stable, then so is S provided the map θ is L-full.

2. If S is left L-stable, then so is R provided the map θ is L-ﬁt.

3. R is SR1 (IC or DF) if and only if S is SR1 (IC or DF).

4. If R is left UG, then so is S. The converse is true provided that for any b ∈ R, lR(b) = lR¯(¯c) for some c ∈ R, where R¯ = R/hxi andr ¯ = r + hxi for any r ∈ R.

Proof. Observe that if R = S[[x]], then we have R = S ⊕ hxi is an ideal extension of S, where J(R) = J(S) ⊕ hxi. (1) and (2). These follow using Theorem 2.4.5 because the map θ is L-full and L-ﬁt by the assumptions. (3) and (4). These follow using Proposition 2.4.6.

We now turn to the polynomial rings:

Proposition 2.5.2. Let L be any left idealtor. For the polynomial ring R = S[x] over the ring S, we have the following facts:

1. If R is SR1, then so is S. The converse need not be true in general.

2. If R is left UG, then so is the ring S.

3. If R is IC, then so is the ring S. The converse need not to be true in general.

4. R is DF if and only if S is DF.

5. If S is left L-stable, then R is not left L-stable in general.

Proof. For the polynomial ring R = S[x] over the ring S, we have R = S⊕hxi is a Dorroh extension. (1). If R is SR1, then so is its factor ring R/hxi using Proposition 2.1.3. Hence, S is SR1 because ∼ 12 S = R/hxi. For the converse, every ﬁeld F ⊆ R has stable range 1 (SR1). But, F [x] always has

12As mentioned earlier, isomorphisms preserve the SR1 condition using Proposition 3.4.8.

34 stable range 2 by Example 1.6 in [73] (see also [108]).13 (2). This follows using Corollary 11 in [95]. (3). If R is IC, then so is S using Corollary 5.4 in [64]. In addition, the converse is not true in general as the ring discussed in Proposition 5.10 [64] is IC, but its polynomial ring is not IC. (4). As mentioned earlier, a unital subring S of a DF ring R is again DF. Hence, if R is DF, then so is S being unital subring of R. Conversely, assume that S is a DF ring. Then, the ring of formal power series W = S[[x]] is DF by (3) of Proposition 2.5.1. Therefore, R is a DF ring because R is a unital subring of W . (5). This follows because the SR1 and the IC conditions do not pass to polynomial rings by (1) and (3) above.

In view of Proposition 2.5.2 we ask:

Question 2.5.3. Let R = S[x] be the polynomial ring over the ring S. Assuming that S is SR1 (IC or left UG), when is R SR1 (IC or left UG)?

In addition, as we have seen above all the ring properties SR1, left UG, IC and DF pass from the polynomial ring R = S[x] to the ring S. So, one may ask:

Question 2.5.4. Can this be generalized to L-stability? That is, assume that the polynomial ring R = S[x] is left L-stable, where L is any left idealtor and S is any ring. Is S left L-stable? When does the converse hold?

2.6 Matrix Rings

We ﬁrst verify that L-stability does not pass to matrix rings in general, and hence it may not be a Morita invariant property of rings. However, we next prove that if a matrix ring over a ring R is left L-stable, then so is R. More generally, we see that if a generalized matrix ring over the rings Ri is left L-stable, then so is each Ri provided certain assumptions hold. Finally, in the last two parts of this section, we show that L-stability passes to special kinds of matrix rings assuming certain conditions; the generalized upper triangular matrix rings and the generalized context-null extension rings.

13Following Bass [9], the SR1 condition can be generalized as follows: we say that an integer n ≥ 1 is in the stable range of a ring R if, whenever r1R + ··· + rn+1R = R with ri ∈ R, then there exist elements x1, . . . , xn ∈ R such n that Σi=1(ri + rn+1xi)R = R, and so the stable range of R is the smallest integer n in the stable range of R [69]. This important notion discussed by many authors in the literature as [32], [33] and [82].

35 2.6.1 L-stability is not a Morita Invariant Property of Rings in General

Recall that a ring-theoretic property p is said to be Morita invariant if and only if, whenever a 14 ring R satisﬁes p, so do eRe (for any full idempotent e ∈ R ) and Mn(R) (for any n ≥ 2). For example, the regularity condition is a Morita invariant property of rings.

It is well known that the SR1 condition passes to corners and matrix rings, and hence it is a Morita invariant property of rings [73]. Thus, we conclude that L-stability is a Morita invariant property of rings for the particular left idealtor S(R) = {L | L is a left ideal of R}. So, it is natural to ask whether or not this will be the case for any left idealtor L. Unfortunately, the answer is dashed by:

Example 2.6.1. The left UG, IC and DF properties do not pass to matrix rings in general, and thus each of them is not a Morita invariant property of rings.

Proof. Shepherdson’s Example [106]. Let S = Z[x11, x12, x21, x22, y11, y12, y21, y22] be the polynomial ring in noncommuting indeterminants xij and yij, and let A denote the ideal of S generated by the following four polynomials:

x11y11 + x12y21 − 1, x11y12 + x12y22, x21y11 + x22y21, x21y12 + x22y22 − 1.

Deﬁne R = S/A, and write aij = xij + A and bij = yij + A for all i and j. Then, the matrices a = [aij] and b = [bij] in M2(R) satisfy ab = 1, but ba 6= 1. Thus, M2(R) is not DF, and hence neither IC nor left UG using Proposition 1.4.1. In addition, R is left UG being a domain, and so it is IC and DF again using Proposition 1.4.1.

Hence, we conclude that:

Proposition 2.6.2. In general, L-stability may not pass to matrix rings, and hence it may not be a Morita invariant property of rings.

So, we ask:

Question 2.6.3. The following two questions:

1. When does L-stability (left UG or IC) pass to matrix rings?

2. When is L-stability (left UG or IC) a Morita invariant property of rings?

14An idempotent element e2 = e in a ring R is said to be full if ReR = R. Many ring-theoretic properties pass to corners eRe for any idempotent e without the fullness assumption.

36 2.6.2 Generalized Matrix Rings

First, we recall the deﬁnition of the generalized matrix rings:

Deﬁnition 2.6.4. Let R1,...,Rn be rings and, whenever i 6= j, let Vij be an Ri-Rj-bimodule.

Moreover, assume that there exists a multiplications VijVji ⊆ Ri for each i, and VijVjk ⊆ Vik when i 6= k, such that

  R1 V12 ...V1n     V21 R2 ...V2n Λ = Mn[Ri,Vij] =   (2.3)  . . .. .   . . . .    Vn1 Vn2 ...Rn is an associative ring using matrix operations. Then, we call the ring Λ a generalized n × n matrix ring over Ri (also known as “generalized Morita context ring”). The prototype example is Λ = End(RM) where M = M1 ⊕ · · · ⊕ Mn, Ri = End(RMi) for each i, and Vij = HomR(Mi,Mj) when i 6= j. For the case n = 2, we get the ring of a Morita context (R, RVS, SWR, S).

As we have seen earlier, L-stability does not pass to matrix rings in general. Thus, we conclude that:

Proposition 2.6.5. Assume that the rings Ri are left L-stable, where i = 1, . . . , n. Then, the generalized matrix ring Λ = Mn[Ri,Vij] is not left L-stable in general.

Hence, the following question is of interest:

Question 2.6.6. When is the generalized matrix ring Mn[Ri,Vij] over the left L-stable rings Ri again left L-stable?

The left idealtor L is said to be nice if any ring isomorphism is L-ﬁt (equivalently, L-full).15 Despite Proposition 2.6.5, we have the following result:

Theorem 2.6.7. Let L be any nice left idealtor, and let R = Mn[Ri,Vij] be a generalized matrix ring over the rings Ri. If R is left L-stable, then so is each Ri provided the following conditions hold for any e ∈ D(R):

1. If X ∈ L(eRe), then RX ∈ L(R).

15As we will see later in Section 3.4, isomorphisms do not preserve L-stability in general. However, if the left idealtor L is nice, then isomorphisms do preserve L-stability.

37 2. One of the following two statements holds:

(a) Every left L-stable ring is DF.

(b) The map θ : R → eRe, deﬁned by θ(r) = ere for all r ∈ R, is a ring morphism.

Proof. Assume that R is left L-stable. Then, each of the corners eiiReii is left L-stable follows by

Theorem 2.2.1 using the conditions (1) and (2), where eii is the matrix unit for each i. As we have ∼ Ri = eiiReii for each i, each Ri is also left L-stable because L is a nice left idealtor by assumption, and hence isomorphisms preserve L-stability, proving the result.

Question 2.6.8. Can the conditions in Theorem 2.6.7 be weakened?

Using Proposition 2.6.5 and Theorem 2.6.7 with n = 2, the following result holds for the ring of a Morita context which generalizes Theorem 29 in [95] to an arbitrary left idealtor L:

RV Corollary 2.6.9. Let Λ = be the ring of a Morita context. If Λ is left L-stable, then WS both S and R are left L-stable provided the conditions in Theorem 2.6.7 hold for Λ. Moreover, the converse need not be true in general.

Although L-stability does not pass to matrix rings, using Theorem 2.6.7 with Ri = R = Vij for each i and j, we have:

Corollary 2.6.10. Let R be any ring, and let n be any ﬁxed positive integer. If the matrix ring

S = Mn(R) is left L-stable, then so is R provided the conditions in Theorem 2.6.7 hold for S.

We discuss below the generalized matrix rings over the particular classes of SR1 rings, left UG rings, IC rings and DF rings; we verify and generalize some known facts for these rings, and get some new results about them:

SR1 rings

As mentioned earlier, the SR1 condition passes to matrix rings [73]. Here, we show that the converse is also true, as is a generalization:

Proposition 2.6.11. If the generalized matrix ring Λ = Mn[Ri,Vij] over the rings Ri is SR1, then so is each Ri. In particular, for any ring R and any ﬁxed positive integer n, the matrix ring Mn(R) is SR1 if and only if R is SR1.

38 Proof. Assume that Λ is SR1. Hence, each of the corner rings eiiReii is SR1 using Corollary 2.2.2, ∼ where eii is the matrix unit for each i. Now, as we have Ri = eiiReii, then it follows that each Ri is SR1, proving the ﬁrst statement. For the last statement, the forward implication follows using the above argument with Ri = R = Vij for each i and j, and the converse follows using our earlier discussion about SR1.

Question 2.6.12. Is the converse of the ﬁrst part of Proposition 2.6.11 true? That is, if each of the rings Ri is SR1, is the generalized matrix ring Λ = Mn[Ri,Vij] SR1?

Left UG rings, IC rings and DF rings

As noticed earlier, the left UG, IC and DF properties do not passes to matrix rings in general. Despite this, the next result shows that the converse is true, it veriﬁes and generalizes part of Proposition 5.7 in [64] and Theorem 29 in [95], and it gives new information for the class of DF rings:

Proposition 2.6.13. Let Λ = Mn[Ri,Vij] be a generalized matrix ring over the rings Ri. If Λ is left UG, IC or DF, then so is each Ri. The converse need not be true in general. In particular, for a ring R and any ﬁxed positive integer n, if the matrix ring S = Mn(R) is left UG, IC or DF, then so is R.

Proof. Assume that Λ is a left UG ring. Hence, each of the corner rings eiiReii is left UG using ∼ Corollary 2.2.2, where eii is the matrix unit for each i. Now, as we have Ri = eiiReii, then it follows that each Ri is also left UG. The converse is not true as Example 2.6.1 veriﬁes. The last statement is clear. Similarly, the result holds for the IC and DF properties.

2.6.3 Generalized Upper Triangular Matrix Rings and Generalized Context- null Extensions

Deﬁnition 2.6.14. If we have Vij = 0 in (2.3) whenever i > j, then Λ = Mn[Ri,Vij] is an upper triangular and is called a generalized upper triangular matrix ring over Ri, and we denote this by Tn[Ri,Vij]. In the case n = 2, we get the usual split-null extension:   R1 V12 T2[Ri,Vij] =  . 0 R2

39 RV r v Remark 2.6.15. If Λ = is the split-null extension and λ = ∈ Λ, then we can write 0 S 0 s λ = δ + η where

r 0 0 v δ = and η = ∈ J(Λ). 0 s 0 0

More generally, if Λ = Tn[Ri,Vij] is a generalized upper triangular matrix ring over the rings Ri, then Λ = S ⊕ A is an ideal extension, where S is the subring of diagonal matrices and A is the ideal of matrices with zero diagonal. Moreover, A ⊆ J(Λ) because A is nilpotent, and we also have n ∼ W = Πi=1Ri = S.

Despite Proposition 2.6.5, the next main result shows that L-stability passes to the generalized upper triangular matrix rings assuming certain conditions. In particular, this gives a generalization of Theorem 36 in [95] to any left idealtor L:

Theorem 2.6.16. Let R = Tn[Ri,Vij] be a generalized upper triangular matrix ring over the rings n Ri. Let W = Πi=1Ri, and consider the decomposition R = S ⊕A, where S is the subring of diagonal matrices in R and A is the ideal of matrices in R with zero diagonal. Deﬁne the map ϕ : R → S by ϕ(s + a) = s for any s ∈ S and any a ∈ A. Let L be any nice left idealtor.16 Then:

1. If R is left L-stable, then so is each Ri provided:

(a) X ∈ L(S) implies X = ϕ(L) for some L ∈ L(R).

n (b) Li ∈ L(Ri), for each i, implies Πi=1Li ∈ L(W ).

2. If each Ri is left L-stable, then so is R provided:

n (a) L ∈ L(W ) implies L = Πi=1Li where Li ∈ L(Ri) for each i. (b) L ∈ L(R) implies ϕ(L) ∈ L(S).

Proof. Notice that, using Remark 2.6.15, we have R = S ⊕ A is an ideal extension where S =∼ W and A ⊆ J(R). So, we use Theorem 2.4.5 and Theorem 2.3.1 to prove both directions as follows: (1). Let R be left L-stable. Then, S is left L-stable follows by Theorem 2.4.5 because ϕ is L-full n ∼ using condition (a) in (1). It follows that W = Πi=1Ri is left L-stable because S = W and L is a nice left idealtor by assumption, and hence isomorphisms preserve L-stability. Therefore, each Ri is left L-stable follows by Theorem 2.3.1 using condition (b) in (1), as required.

16As mentioned earlier, isomorphisms do not preserve L-stability in general. However, they do so for any nice left idealtor L we will see in Section 3.4.

40 n (2). Assume now that each Ri is left L-stable. Hence, we have W = Πi=1Ri is left L-stable by Theorem 2.3.1 using condition (a) in (2). Since S =∼ W , it follows that S is also left L-stable because L is a nice left idealtor by assumption. Therefore, R is left L-stable by Theorem 2.4.5 because ϕ is L-ﬁt using condition (b) in (2), proving (2).

With n = 2, we immediately get:

RV Corollary 2.6.17. Consider the split-null extension Λ = , and let L be any left idealtor. 0 S Assume that the conditions in Theorem 2.6.16 hold for Λ, R and S. Then, Λ is left L-stable if and only if both S and R are left L-stable.

If we choose Ri = R = Vij for each i and j, we also get:

Corollary 2.6.18. Let U be an upper triangular matrix ring over a ring R, and let L be any left idealtor. Assume that the conditions in Theorem 2.6.16 hold for U and R. Then, U is left L-stable if and only if R is left L-stable.

Finally, we ask:

Question 2.6.19. Can the conditions in Theorem 2.6.16 and its corollaries be weakened?

SR1, left UG, IC and DF Properties

Here, we apply the results discussed above to the particular classes: SR1 rings, left UG rings, IC rings and DF rings to get some well-known facts and new results about these rings.

Proposition 2.6.20. Let R = Tn[Ri,Vij] be a generalized upper triangular matrix ring over the rings Ri. Then:

1. R is SR1 (IC or DF) if and only if each Ri is SR1 (IC or DF).

2. R is left UG if and only if each Ri is left UG.

Proof. (1). For the SR1 rings, the result follows using Theorem 2.6.16 applied to the left idealtor S(R) = {L | L is a left ideal of R} because its conditions hold in this case by Proposition 2.4.6 and Corollary 2.3.2. Similarly for the IC rings and DF rings.

(2). If R is left UG, so is each Ri using (1) of Theorem 2.6.16 applied to the left idealtor A(R) = {l(a) | a ∈ R} (the conditions in (1) hold in this case by Proposition 2.4.6 and Corollary 2.3.2). The other implication follows by Theorem 36 in [95].

41 Corollary 2.6.21. Let R be an upper triangular matrix ring over the ring S. Then, R is SR1 (left UG, IC or DF) if and only if S is SR1 (left UG, IC or DF).17

We end this section with the following observation:

Remark 2.6.22. The generalized matrix ring Λ = Mn[Ri,Vij] over the rings R1, ··· ,Rn in (2.3) is said to be a generalized context-null extension of the rings Ri, denoted by CNn[Ri,Vij], if

VpjVjq = 0 whenever j 6= p or j 6= q. One easily veriﬁes that all the results mentioned above for the generalized upper triangular matrix rings hold for the generalized context-null extensions.

17Notice that the result proved in [95] for the left UG condition (Theorem 36).

42 Chapter 3

Special Classes of Left Idealtors

In this chapter, we investigate three important special classes of left idealtors; the normal left idealtors, the closed left idealtors and the nice left idealtors. These left idealtors are important because some of our major results hold within these classes. Actually, we have already proved a few main results within some of them (Theorems 2.6.7 and 2.6.16 required that the left idealtor L is nice).

3.1 Normal Left Idealtors

We start by proving the following preparation lemma:

Lemma 3.1.1. Let L be any left idealtor. Then, the following two statements are equivalent for any ring R:

1. Lu ∈ L(R) for any L ∈ L(R) and any u ∈ U(R).

2. uLu−1 ∈ L(R) for any L ∈ L(R) and any u ∈ U(R).

Proof. Observe that uL = L for any u ∈ U(R) and any left ideal L of R. (1) ⇒ (2). If L ∈ L(R), then we have uLu−1 = (uL)u−1 = Lu−1 ∈ L(R) by (1), as required. (2) ⇒ (1). Assume that L ∈ L(R) and write M = u−1Lu ∈ L(R) by (2). It follows that Lu = uM = M ∈ L(R), proving (1).

Deﬁnition 3.1.2. A left idealtor L is said to be normal if the conditions in Lemma 3.1.1 are satisﬁed.

43 Theorem 3.1.3. The classes {SR1}, {left UG}, {IC} and {DF} are aﬀorded, respectively, by the following normal left idealtors:

1. S(R) = {L | L is left ideal of R}.

2. A(R) = {l(b) | b ∈ R}.

3. D(R) = {Re | e2 = e ∈ R}.

4. J (R) = {L ⊆ J(R) | L is left ideal of R}.

Proof. We check that each of these left idealtors is normal as follows: (1). SR1: S is normal because if u is any unit of a ring R and L ∈ S(R), then we have Lu ∈ S(R) because Lu is a left ideal of R, as required. (2). Left UG: If u is any unit in a ring R and L = l(b) ∈ A(R), then we have:

uLu−1 = ul(b)u−1 = l(ubu−1) ∈ A(R).

(3). IC: If L ∈ D(R), then L = Re, e2 = e. So, for any u ∈ U(R), we have

Lu = (Re)u = Rf ∈ D(R) where f = u−1eu ∈ D(R).

(4). DF: J is clearly normal because J(R) C R for any ring R.

It follows that:

Proposition 3.1.4. If an element a in a ring R is SR1, left UG, IC or DF, then so are ua and au for any u ∈ U(R).

Proof. These classes are aﬀorded by the left idealtors S, A, D and J respectively. Use Theorem 3.1.3, and apply the Translation Lemma 1.4.12.

3.2 Closed Left Idealtors

Deﬁnition 3.2.1. Let L be any left idealtor. Deﬁne the new left idealtor L¯ for each ring R by

L¯(R) = {M | M is a left ideal of R and M =∼ L for some L ∈ L(R)}.

We call L¯ the closure of the left idealtor L, and say that L is closed if L¯ = L.

44 The topological terminology is justiﬁed by the following Lemma:

Lemma 3.2.2. The following statements are true for any left idealtor L:

1. L(R) ⊆ L¯(R) for each ring R.

2. L¯ = L¯.

Proof. (1). For any L ∈ L(R), we have L =∼ L and so L ∈ L¯(R), proving (1). (2). Applying (1) to L¯ yields L¯(R) ⊆ L¯(R) for each ring R. Now, let X ∈ L¯(R), so there exists M ∈ L¯(R) such that X =∼ M. Then, in turn, let M =∼ L ∈ L(R). Thus, X =∼ M =∼ L ∈ L(R) which implies X ∈ L¯(R). Therefore, we also have L¯(R) ⊆ L¯(R), so L¯(R) = L¯(R) for each ring R.

Examples of closed left idealtors include:

Example 3.2.3. 1. S(R) = {L | L is a left ideal of R} and P(R) = {Rb | b ∈ R} that aﬀord the class of SR1 rings.

2. T (R) = {0} that aﬀords the class of DF rings.

Proof. (1). S is clearly closed by the deﬁnition. Moreover, P is closed because if N is any left ideal of R such that N =∼ Rb for some b ∈ R, then N = Rϕ(b) where ϕ : Rb → N is an isomorphism. (2). Let N be any left ideal of a ring R such that N =∼ L ∈ T (R). Then, we have N =∼ 0, and so N = 0 ∈ T (R) which implies that T is closed, as desired.

On the other hand, not all left idealtors are closed:

Example 3.2.4. Each of the following left idealtors is not closed:

1. C(R) = {R} that aﬀords the class of all rings.

2. The left idealtors that aﬀord the class of DF rings mentioned in Example 1.2.12;

J (R), J1(R), J2(R), J3(R), and J4(R).

3. A(R) = {l(a) | a ∈ R} that aﬀords the class of left UG rings.

4. D(R) = {Re | e2 = e ∈ R} that aﬀords the class of IC rings.

∼ Proof. (1). Consider the ring of integers Z. Then, we have L = Z ∈ C(Z) = {Z} and M = Z2 = L via the isomerism k 7→ k2, but M/∈ C(Z). Hence, C is not closed. FF (2). Let R = , where F is any ﬁeld. Then, the two left ideals 0 F

45 F 0 0 F M = and J(R) = 0 0 0 0 are isomorphic. On the other hand, J(R) is in each of these left idealtors, but M is in none. Therefore, none of them is closed.

(3). For the ring of integers Z, we have A(Z) = {Z, 0} using Example 1.2.16. Therefore, A is not closed with a similar argument used in (1).

(4). Consider again the ring of integers Z. Then, D(Z) = {Z, 0} using Example 1.2.16. Therefore, D is not closed with a similar argument used in (1).

Remark 3.2.5. We may have two left idealtors that afford the same class of rings, where one of them is closed and the other is not. Indeed, the left idealtors in (2) of Example 3.2.3 and (2) of Example 3.2.4 all afford the class of DF rings, but the first is closed and the others are not.

As noticed earlier, there are closed left idealtors that aﬀord the classes of SR1 rings and DF rings. On the other hand, the left idealtors

A(R) = {l(a) | a ∈ R} and D(R) = {Re | e2 = e ∈ R} that aﬀord the classes of left UG rings and IC rings, respectively, are not closed. So, we ask:

Question 3.2.6. Does there exist a closed left idealtor aﬀording the class of IC rings? The class of left UG rings?

We end this part proving the following result that compares the class of closed left idealtors and the class of normal left idealtors as follows:

Proposition 3.2.7. Let L be any left idealtor. If L is closed, then it is normal. The converse need not be true in general.

Proof. Assume that L is closed, and let R be any ring. If L ∈ L(R), then we have L =∼ Lu for any u ∈ U(R) via the isomorphism l 7→ lu. Thus, Lu ∈ L(R) because L is closed, and hence L is normal, as required. For the converse, the left idealtor D(R) = {Re | e2 = e ∈ R} is normal using Theorem 3.1.3, but it is not closed using (4) of Example 3.2.4.

3.3 Products of left L-stable elements

In this section, we ﬁrst prove one of the major results in the thesis which asserts that, for any normal left idealtor L, the set of all left L-stable elements in any ring R, denoted by SL(R), is a

46 multiplicative submonoid of R. This result and its consequences generalize some well-known facts for the SR1 property, and also give us new facts for the left UG, IC and DF properties. For example we answer an open question in [95]; whether or not the UG property closed under multiplication.

Finally, we show that the set SL(R) is not an additive submonoid of R in general.

Recall that, in the Translation Lemma 1.4.12, we have shown if a ∈ SL(R) and L is a normal left idealtor, then so are au and ua for any unit u in any ring R. In the next fundamental result, we extend this fact to any two elements in SL(R). In particular, this generalizes the same result for the SR1 case in [35].

Theorem 3.3.1. Let L be any normal left idealtor, and let R be any ring. If a, b ∈ SL(R), then so is their product ab.

Proof. Let d and a be two left L-stable elements in R, and assume that Rda+L = R with L ∈ L(R), say rda + b = 1 where r ∈ R and b ∈ L. Then, Ra + L = R, and so a − u ∈ L for some u ∈ U(R) because a is left L-stable by assumption, that is, a = u + c for some c ∈ L . It follows that, 1 = rda + b = rd(u + c) + b, and so rdu + (rdc + b) = 1, which implies that

rdu + g = 1 where g = rdc + b ∈ L because c, b ∈ L.

Multiplying on the left by u, and then on the right by u−1 to obtain:

urd + ugu−1 = 1, from which Rd + uLu−1 = R.

Now, as L is normal, we have uLu−1 ∈ L(R). Thus, d − v ∈ uLu−1 for some v ∈ U(R) because d is left L-stable by assumption, say d − v = uhu−1 where h ∈ L. So, du − vu = uh, but then d(a − c) − vu = uh because a − c = u, that is, da − (dc + uh) = vu. Finally, as vu ∈ U(R) and dc + uh ∈ L, it follows that da is left L-stable, as required.

This result also holds for the class of closed left idealtors using Proposition 3.2.7. So, we conclude the following:

Corollary 3.3.2. Let L be any closed left idealtor, and let R be any ring. If a, b ∈ SL(R), then so is their product ab.

As a consequence of Theorem 3.3.1, we generalize Proposition 3.1.4, we get Lemma 17 in [35], we answer the (open) Question (1) in [95], and we get a new fact for the classes of IC rings and DF rings as follows:

47 Corollary 3.3.3. Let R be any ring. If a, b ∈ SS (R)[SA(R), SD(R) or SJ (R)], then so is their product ab.

Proof. The SR1 rings, left UG rings, IC rings and DF rings are aﬀorded, respectively, by the left idealtors

P(R) = {Rb | b ∈ R}, A(R) = {l(a) | a ∈ R}, D(R) = {Re | e2 = e ∈ R} and T (R) = {0}.

Moreover, these idealtors are normal using Theorem 3.1.3, and hence the result now follows using Theorem 3.3.1.

In view of Theorem 3.3.1, we ask:

Question 3.3.4. Let L be any left idealtor, and let R be a ring. Assume that ab ∈ SL(R). Is a left L-stable? Is b left L-stable?

Next, we prove the following result:

Proposition 3.3.5. Let L be a normal (or closed) left idealtor, and let R be any ring. Then,

1. The set SL(R) is a multiplicative submonoid of R.

2. SL(R) contains ureg(R), J(R), U(R) and D(R). Moreover, it contains Zr(R) when r(L) 6= 0 for each R 6= L ∈ L(R), in particular, when R is left Kasch.

3. SL(R) = R if and only if R is left L-stable.

4. Deﬁne the map ϕ : SL(R) → SL[R/J(R)] by ϕ(a) = a + J(R), and assume that J(R) is L-ﬁt. Then, ϕ is an onto monoid morphism with ϕ(a) = ϕ(b) if and only if a − b ∈ J(R).

Proof. (1). SL(R) is a multiplicative submonoid of R follows using Theorem 3.3.1 and the fact that

1 ∈ SL(R). (2). This follows using Theorem 1.4.13. (3). This is clear using the deﬁnitions. (4). Write R¯ = R/J(R) andr ¯ = r + J(R) for any r ∈ R. Clearly ϕ(ab) = ab =a ¯¯b = ϕ(a)ϕ(b) ¯ for all a, b ∈ SL(R). Moreover, ϕ(1) = 1,¯ and ϕ(a) = ϕ(b) if and only ifa ¯ = b, that is, a−b ∈ J(R).

So, it remains to show that ϕ is onto. Now, ifa ¯ ∈ stabL(R¯) with a ∈ R, thena ¯ is left L-stable in R¯. Hence, a is left L-stable in R using Corollary 2.1.7 because J(R) is L-ﬁt by assumption.

In particular, we get Proposition 18 in [35] and new facts that have not been discussed before for the classes of left UG rings, IC rings and DF rings as follows:

48 Corollary 3.3.6. Let R be a ring. Then:

1. Proposition 3.3.5 holds for SS (R), SD(R) and SJ (R).

2. The ﬁrst three statements in Proposition 3.3.5 hold for SA(R), and the fourth holds provided

that b ∈ R implies lR(b) = lR¯(¯c) for some c ∈ R.

Proof. (1). The SR1 rings, IC rings and DF rings are afforded, respectively, by left idealtors S(R) = {L | L is a left ideal of R}, D(R) = {Re | e2 = e ∈ R} and T (R) = {0} which are normal using Theorem 3.1.3. In addition, J(R) is S-fit, D-fit and T -fit using Proposition 2.1.8. Therefore, the result follows after applying Proposition 3.3.5. (2). The left idealtor A(R) = {l(a) | a ∈ R} that affords the class of left UG rings is normal using Theorem 3.1.3. On the other hand, J(R) is A-fit by Proposition 2.1.8 using the assumption. Hence, the result follows again using Proposition 3.3.5.

Is SL(R) an Additive Submonoid of R?

In the previous part, we have shown that the set SL(R) is a multiplicative submonoid of R for any normal (closed) left idealtor L. So, a natural question to ask whether or not this will be the case for addition. That is:

For a left idealtor L, if a and b are in SL(R), is their sum a + b also in SL(R)?

Unfortunately the answer is dashed by:

1 Example 3.3.7. For the ring of integers Z, we have SS (Z) = {0, 1, −1}.

Proof. Clearly we have {0, 1, −1} ⊆ SS (Z). Suppose k ∈ SS (Z)\{0, 1, −1}. If p is any prime with 2 p - k, then Zk + Zp = Z. As k ∈ SS (Z), we have p | (k − 1) or p | (k + 1). It follows that p | (k − 1), a contradiction because there are inﬁnitely many such primes p. Hence, SS (Z) ⊆ {0, 1, −1}, which ends the proof.

So, we ask:

Question 3.3.8. Let L be any left idealtor. When is the set SL(R) an additive submonoid of R?

Observe that if R is left L-stable, then R = SL(R) is a subring of R. Moreover, if the answer to Question 3.3.8 is yes assuming certain condition(s). Then, in this case we conclude that the

1 Notice that Z is not SR1 using Example 1.2.16.

49 set SL(R) is an additive submonoid of R where “0” is its additive identity, and hence using (1) of Proposition 3.3.5 we conclude that the set SL(R) is a subring of R for any normal (closed) left idealtor L.

Finally, notice that the Shift Lemma 2.4.4 gives us a partial answer to Question 3.3.8 because it asserts that:

2 For a ring R, SL(R) + J(R) ⊆ SL(R) for any left idealtor L.

3.4 Nice Left Idealtors

Theorem 2.1.6 asserts that any onto ring morphism ϕ : R → S preserves L-stability provided that ϕ is both L-ﬁt and L-full. So, it is natural to ask whether or not these assumptions can be dropped and L-stability is preserved in the case that ϕ is a ring isomorphism. In this section, we discuss this and provide examples to show that this is not the case in general. On the other hand, we introduce a new class of left idealtors; the class of nice left idealtors, in which L-stability is preserved under any ring isomorphism.

First, in the next example we show that isomorphisms in general are not necessarily L-ﬁt and L-full for a given left idealtor L:

Example 3.4.1. Fix a ring Ro with J(Ro) 6= 0. Deﬁne, the left idealtor L for each ring R as follows:   {J(Ro)} , if R = Ro L(R) =  {0} , if R 6= Ro

Write S = Ro × {0}, and deﬁne the isomorphism ϕ : S → Ro by (r, 0)ϕ = r for any r ∈ Ro. Then, ϕ is neither L-ﬁt nor L-full.

Proof. If L ∈ L(S), then by deﬁnition we have L = 0. Thus, ϕ(L) = ϕ(0) = 0 ∈/ L(Ro), and hence

ϕ is not L-ﬁt. In addition, if X ∈ L(Ro), then by deﬁnition we have X = J(Ro) 6= 0. On the other hand, L = 0 is the only element in L(S) and ϕ(L) = ϕ(0) = 0 6= J(Ro) = X which implies that ϕ is not L-full, as required.

Lemma 3.4.2. Let Mω(D) = end(DV ) where D is a division ring and DV is a vector space with basis {v0, v1, ···}. Then, Mω(D) is not directly ﬁnite.

2 Recall that for a ring R, we always have J(R) ⊆ SL(R) for any left idealtor L using Theorem 1.4.13.

50 Proof. Deﬁne the shift operator σ ∈ Mω(D) by viσ = vi+1 for all i; and the co-shift operator

τ ∈ Mω(D) by voτ = 0 and viτ = vi−1 for all i ≥ 1. Then, we have στ = 1V , but τσ 6= 1V because voτσ = 0. So, Mω(D) is not a DF ring as desired.

We are now ready to present the second example which veriﬁes that isomorphisms do not preserve L-stability in general as follows:

∼ Example 3.4.3. Let D be a division ring, and write E = Mω(D) and S = E × {0}. Then, E = S as rings via ε 7→ (ε, 0) for any ε ∈ E. Deﬁne, the left idealtor L for each ring R as follows:

  {E} , if R = E L(R) =  {0} , if R 6= E

Then, E is left L-stable, but S is not left L-stable.

Proof. To see that E is left L-stable, let Eε+L = E with ε ∈ E and L ∈ L(E). Since L(E) = {E}, we have L = E and so ε − 1 ∈ L. Now, suppose that S is left L-stable. We have L(S) = {0} as S 6= E. Hence, S must be a DF ring by Theorem 1.2.8. But, this is a contradiction because S =∼ E is not a DF ring using Lemma 3.4.2 and (4) of Proposition 3.4.8 which will be proved shortly. Therefore, S is not a left L-stable ring, as required.

We prove the following key result:

Lemma 3.4.4. Let L be any left idealtor.

ρ τ 1. Let R −→S −→R be ring morphisms with τ ◦ ρ = 1R. Then, we have:

(a) ρ is L-ﬁt implies τ is L-full.

(b) ρ is L-full implies τ is L-ﬁt.

2. If R −→σ S is a ring isomorphism, then the following statements hold:

(a) σ is L-ﬁt if and only if σ−1 is L-full.

(b) σ is L-full if and only if σ−1 is L-ﬁt.

3. Every ring isomorphism is L-ﬁt if and only if every ring isomorphism is L-full.

Proof. (1). Assume that L ∈ L(R). Then, we have L = τ[ρ(L)], and ρ(L) ∈ L(S) because ρ is L-ﬁt, proving (a). For (b), let X ∈ L(S). As ρ is L-full we have X = ρ(L) for some L ∈ L(R). Then, we have τ(X) = τ[ρ(L)] = L ∈ L(R), as required.

51 (2). First, we notice that (a) implies (b) by σ 7→ σ−1. But, (a) follows using (1) because:

using 1(a) using 1(b) σ is L-ﬁt =⇒ σ−1 is L-full =⇒ σ is L-ﬁt.

(3). Assume first that every ring isomorphism is L-fit. If σ is a ring isomorphism, then σ−1 is L-fit. So, σ = (σ−1)−1 is L-full by 2(a). The proof of the converse is similar.

In view of Lemma 3.4.4, we introduce a new class of left idealtors as follows:

Deﬁnition 3.4.5. Call a left idealtor L nice if every ring isomorphism is L-full (equivalently, L-ﬁt).

Hence, we conclude that ring isomorphisms preserve L-stability in this case as follows:

Theorem 3.4.6. Let L be a nice left idealtor. If R =∼ S as rings, then R is left L-stable if and only if S is left L-stable.

Proof. Let σ : R → S be a ring isomorphism. Since L is nice, then σ is L-full and L-ﬁt. Therefore, R is left L-stable if and only if S is left L-stable follows using Theorem 2.1.6, as required.

As we have noticed above, in Example 3.4.1 and Example 3.4.3, that not all left idealtors are nice. However, the next example veriﬁes that nice left idealtors do exist:

Example 3.4.7. The following left idealtors are nice:

1. S(R) = {L | L is a left ideal of R}.

2. P(R) = {Ra | a ∈ R}.

3. A(R) = {l(a) | a ∈ R}.

4. D(R) = {Re | e2 = e ∈ R}.

5. T (R) = {0}

Proof. Let ϕ : R → S be any ring isomorphism. To prove the result, in each case we verify that the map ϕ satisfies either the fullness assumption or the fitness assumption as follows: (1). If X ∈ S(S) is any left ideal of S, then L = ϕ−1(X) is a left ideal of R with X = ϕ(L). Therefore, there exists L ∈ S(R) such that X = ϕ(L), and hence ϕ is S-full, as required. (2). If N ∈ P(R), then N = Ra for some a ∈ R. Now, ϕ(Ra) = Sϕ(a) ∈ P(S), and hence ϕ is P-fit, as desired.

52 (3). We observe that ϕ[l(r)] = l[ϕ(r)] for any r ∈ R. Indeed, if x ∈ ϕ[l(r)], then x = ϕ(a) with ar = 0. Thus, ϕ(a)ϕ(r) = 0, and hence x = ϕ(a) ∈ l[ϕ(r)]. On the other hand, if x ∈ l[ϕ(r)], then xϕ(r) = 0. So, ϕ−1(x)r = 0, which implies that ϕ−1(x) ∈ l(r) and hence x ∈ ϕ[l(r)], as required. Now, If X ∈ A(S), then X = l(s) for some s ∈ S. But s = ϕ(r) for some r ∈ R because ϕ is onto. Thus, X = l[ϕ(r)] = ϕ[l(r)], that is, there exists L = l(r) ∈ A(R) such that X = ϕ(L), and hence ϕ is A-full, proving (2). (4). The map ϕ is clearly D-ﬁt because if e ∈ D(R), then we always have ϕ(Re) = Sϕ(e) where ϕ(e) ∈ D(S). Indeed, if x ∈ ϕ(Re), then x = ϕ(re) = ϕ(r)ϕ(e) ∈ Sϕ(e). On the other hand, if x ∈ Sϕ(e), then x = sϕ(e) = ϕ(r)ϕ(e) for some r ∈ R because ϕ is onto. Hence, x = ϕ(re) ∈ ϕ(Re). The last statement is clear. (5). ϕ is clearly T -ﬁt because ϕ[0 ∈ T (R)] = 0 ∈ T (S).

Proposition 3.4.8. Let ϕ : R → S be any ring isomorphism. Then, R is SR1 (left UG, IC or DF) if and only if S is SR1 (left UG, IC or DF).

Proof. The SR1 rings are aﬀorded by the left idealtor S(R) = {L | L is a left ideal of R} which is nice using Example 3.4.7. Hence, R is SR1 if and only if S is SR1 follows using Theorem 3.4.6. With a similar argument, the result also follows for the left UG rings, IC rings and DF rings since they are aﬀorded, respectively, by the nice left idealtors A(R) = {l(a) | a ∈ R}, D(R) = {Re | e2 = e ∈ R} and T (R) = {0}.

Finally, notice that within the class of nice left idealtors we have already proved some major results earlier. Namely, for the matrix rings, Theorem 2.6.7 and Theorem 2.6.16 have been proven assuming that the left idealtor L is nice. In addition, the fact that the SR1, left UG, IC and DF properties are preserved by isomorphisms has been also used in Example 1.4.22, Proposition 2.4.6, Proposition 2.5.2, Proposition 2.6.11, Proposition 2.6.13 and Example 3.4.3.

53 Part II

Left Quasi-duo Rings

54 Chapter 4

Examples and Basic Results

A ring R is called left duo if every left ideal is an ideal of R [46]. In addition, a ring R is called weakly left duo if for each a in R, there exists a positive integer n = n(a), depending on a, such that the left ideal Ran is an ideal of R [115].1 Our interest is in a much larger class of rings; the class of left quasi-duo rings, that is, rings where every maximal left ideal is an ideal [116].2 In this second part of the thesis, we study this large class of rings, prove some interesting results, and give many natural characterizations for them that have passed unnoticed since they have been introduced in 1995 by Yu.

4.1 Motivation

Left duo rings, weakly left duo rings, commutative rings and local rings are examples of left quasi- duo rings. The left quasi-duo rings are related to important questions. First, in 1995, Yu proved that if a ring R is left quasi-duo, then Bass’s conjecture [8] holds; which asserts that a ring R is left perfect3 if it has no inﬁnite orthogonal sets of idempotents (that is, R is I-ﬁnite) and every left R-module has a maximal submodule.4

The study of this class of rings was continued in the work of other authors in [22], [57], [67], [77]

1These rings studied and investigated by many authors; see for example [37], [14], [25], [56], [66], [78] and [114]. 2 Recall that a left ideal L of a ring R is said to be a maximal left ideal if, whenever we have RL⊆RN⊆RR, then either N = L or N = R; equivalently, the left R-module R/L is simple. 3A ring R is called left perfect if R/J(R) is semisimple and every non-zero left R-module has a maximal submodule. 4The fact that this is true for commutative rings was proved by Hamsher [54], and that this is not true in general was veriﬁed by examples of Cozzens [38] and Koifman [68]. In addition, this was proved for left duo rings by Chandran in [24], and for weakly left duo rings by Xue [113].

55 and [104]. In addition, in 2002, Huh-Jang-Kim-Lee [58] showed that if a polynomial ring R[x] is left quasi-duo, then the K¨otheconjecture holds for R, that is,

J(R[x]) = N(R)[x] where N(R) = {a ∈ R | a is nilpotent} is an ideal of R.

This built on earlier work of Kim-Kim-Jang [65] in 1999 who studied the relationship between the left quasi-duo rings and the class of 2-primal rings; that is, rings where their prime radical consists of the nilpotent elements.5

Finally, in 2005, Lam-Dugas [75] showed that a ring R is left quasi-duo if and only if it satisﬁes the left unimodularity condition:

r1R + ··· + rnR = R with ri ∈ R, implies Rr1 + ··· + Rrn = R.

They also concluded that if R is a left and right quasi-duo ring and has stable range at most n, then any corner ring eRe with e2 = e ∈ R (more generally, every semisplit generalized corner6 of R) also has stable range at most n..

In this chapter, we ﬁrst introduce the notion of module rings, and get a characterization of the left quasi-duo rings in terms of their simple left modules. Next, we give a short survey of the basic results on left quasi-duo rings and give new alternative simpler proofs for most of them. After that, we give a compact proof of a characterization of left quasi-duo rings in terms of their very semisimple modules [90]. Finally, we study the generalized upper triangular matrix rings over left quasi-duo rings and get a generalization of Proposition 2.1 in [116].

In the next chapter, we first define a new notion for rings; the left width, and prove one of the major results in this thesis; the Width Theorem. Moreover, we use this result to get a characterization of left quasi-duo rings that have a finite number of non-isomorphic simple left modules. This provides us with a partial answer to the open question; whether every left quasi-duo ring is right quasi-duo. Next, we give a characterization of left quasi-duo rings with no infinite set of orthogonal idempotents. Finally, the last section gives a connection between the two main parts of the thesis; L-stability in rings and left quasi-duo rings. We first introduce a new left idealtor called the left-max idealtor, we also introduce its associated rings which will be called the left-max stable rings. We prove some basic results for both, where we also show that the class of left quasi-duo rings is not affordable.

5These rings have been studied intensely; see for example [59] and [10]. 6For more information about these corners; see [6] and [74].

56 4.2 Module Rings and a “Simple-Modules” Characterization of Left Quasi-duo Rings

Recall that if M is a principal left module over a ring R, then M = Rm for some nonzero element m ∈ M and M =∼ R/l(m) as left R-modules. In this case, we have:

l(M) = {r ∈ R | r[R/l(m)] = 0} = {r ∈ R | rR ⊆ l(m)}, which is the largest two sided ideal of R that is contained in the left ideal l(m). So, it is a natural question to ask; when is the above mentioned isomorphism a ring isomorphism?

The answer to this question is related to the following fundamental fact:

Theorem 4.2.1. For any ring R, the following conditions are equivalent for a principal left R- module Rm:

1. l(m) C R.

2. am = 0 with a ∈ R, implies aRm = 0.

3. l(m) = l(Rm).

In this case, we have: end(Rm) =∼ R/l(m) as rings. (4.1)

Proof. (1) ⇒ (2). Assume that am = 0, then a ∈ l(m). Hence, we have aR ⊆ l(m) by (1), and so (aR)m = 0, as required. (2) ⇒ (3). As discussed above, we always have l(Rm) ⊆ l(m). On the other hand, if a ∈ l(m), then (aR)m = 0 using (2), and so a ∈ l(Rm) which implies that l(m) ⊆ l(Rm), proving (3). (3) ⇒ (1). Clearly, l(m) is a left ideal. Now, if a ∈ l(m), then a ∈ l(Rm) using (3), and so 0 = a(Rm) = (aR)m, that is, aR ⊆ l(m), proving that l(m) is a right ideal. To prove the last statement, let a ∈ R and deﬁne

ϕa: Rm → Rm by (rm)ϕa = ram for each r ∈ R.

This is a well-deﬁned map using (2) above, and it is clearly R-linear. With this in mind, deﬁne the map

θ : R → end(Rm) by θ(a) = ϕa for each a ∈ R.

57 It is a routine veriﬁcation that θ is a ring morphism with kernel l(m), so it remains to show that θ is onto. If β ∈ end(Rm), let mβ = bm with b ∈ R. Hence, (rm)β = r(mβ) = r(bm) = (rm)ϕb for all r ∈ R, and so β = ϕb = θ(b), as required.

Assume now that the left R-module Rm satisﬁes the conditions in Theorem 4.2.1, and deﬁne the multiplication on Rm by:

(am)(bm) = (ab)m for any a, b ∈ R.

Then, this is a well deﬁned multiplication. Indeed, if am = a1m and bm = b1m, and using (1) in Theorem 4.2.1, we have

(ab − a1b1)m = [a(b − b1) + (a − a1)b1]m ∈ al(m)m + l(m)b1m = 0.

It follows that Rm is a ring (with unity m) via this multiplication.

Hence,

Deﬁnition 4.2.2. We shall write Rm = ring(Rm) and call it a module ring. In addition, in this case have: R/l(m) =∼ ring(Rm) as rings. (4.2)

The next result and its consequence, although very straightforward, will be used repeatedly in what follows:

Proposition 4.2.3. The following statements are equivalent for the principal left R-module Rm, where l(m) C R:

1. R/l(m) =∼ ring(Rm) =∼ end(Rm) is a division ring.

2. l(m) = l(Rm) is a maximal left ideal of R.

3. Rm is a simple left R-module.7

In particular, these statements are equivalent for any left quasi-duo ring R.

Proof. The last statement is clear since in this case we have l(m) is an ideal of any left quasi-duo ring R.

7Recall that a non-zero left R-module N is said to be simple if the only submodules of N are 0 and N. This class of left modules will play a major role in this second part of the thesis.

58 (1) ⇒ (2). l(m) is a maximal left ideal of R follows using the Correspondence Theorem because R/l(m) is a division ring by (1). In addition, we have l(m) = l(Rm) using Theorem 4.2.1 because l(m) C R by assumption. (2) ⇒ (3). Assume that l(m) is a maximal left ideal of R. Then, Rm is a simple left R-module because Rm =∼ R/l(m). ∼ ∼ (3) ⇒ (1). As l(m) C R by assumption, we have R/l(m) = ring(Rm) = end(Rm) as rings using 4.2 and 4.1. Now, since Rm is a simple left R-module, it follows that end(Rm) is a division ring by Schur’s Lemma, proving (3).

Using Proposition 4.2.3, we get a characterization of the left quasi-duo rings in terms of their simple left modules as follows:

Corollary 4.2.4. The following two statements are equivalent for any ring R:

1. R is left quasi-duo.

max 2. If K is a simple left R-module, then l(K) = l(k) ⊆ RR is an ideal of R for any 0 6= k ∈ K.

Proof. (1) ⇒ (2). Assume that R is a left quasi-duo ring, and let K be a simple left R-module. max ∼ Then, for any 0 6= k ∈ K, we have Rk = K and l(k) ⊆ RR because R/l(k) = K is simple. Hence, l(k) is an ideal of R because R is a left quasi-duo ring by (1). In addition, l(K) = l(k) follows using Proposition 4.2.3, as required. (2) ⇒ (1). This follows since every maximal left ideal M of R arises as M = l(k) for some nonzero element k in a simple left R-module K, and M is an ideal of R using (2) which implies that R is a left quasi-duo ring, as required.

4.3 Examples and Elementary Properties

In this section, for completeness, we survey the basic properties proved in the literature for the class of left quasi-duo rings that will be used later; many of which appear in [116], [65], [67], [58] and [75]. For almost of them, we give new alternative simpler and shorter proofs than used before.

Deﬁnition 4.3.1. A maximal left ideal of a ring R that is an ideal will be called left-max.

Remark 4.3.2. Every conclusion for left quasi-duo has an obvious “right” version. So, from now on we will conﬁne ourselves to left quasi-duo rings.

59 First, we give a compact proof of the following element-wise characterization of Lam-Dugas [75] for the left quasi-duo rings in terms of the “left unimodularity” condition as follows:8

Lemma 4.3.3. The following statements are equivalent for a ring R:

1. R is left quasi-duo.

2. a1R + ··· + anR = R implies Ra1 + ··· + Ran = R, where ai ∈ R for each i.

3. aR + bR = R implies Ra + Rb = R for all a, b ∈ R.

4. Ra + R(1 − ab) = R for all a, b ∈ R.

Proof. (1) ⇒ (2). Let a1R + ··· + anR = R. If Ra1 + ··· + Ran 6= R, then Ra1 + ··· + Ran ⊆ A for max some A ⊆ RR. But, A is an ideal of R by (1), and so R = a1R + ··· + anR ⊆ A, a contradiction. (2) ⇒ (3). This is clear. (3) ⇒ (4). Clearly aR + (1 − ab)R = R, and so Ra + R(1 − ab) = R using (3), as required. (4) ⇒ (1). Let A be a maximal left ideal of R. Assume that A is not an ideal of R, and let

As * A for some s ∈ R. Hence, As + A = R because A is maximal, say 1 = as + b with a, b ∈ A. But then (4) gives R = Ra + R(1 − as) = Ra + Rb ⊆ A, a contradiction.

Strongly regular rings [107], commutative rings and division rings are left quasi-duo. In addition, any local ring R is left quasi-duo because the only maximal left ideal in R is J(R) which is an ideal. Moreover, Burgess-Stephenson [15] showed that an exchange ring with all idempotents central is quasi-duo. Examples of left quasi-duo rings also include:

DD Example 4.3.4. Consider the ring R = , where D is a division ring. Then: 0 D

1. R is left quasi-duo.

2. R is not abelian.

3. R is neither left duo nor weakly left duo.

Proof. (1). This follows since the only two maximal left ideals of R are

DD 0 D M = and N = , 0 0 0 D

8 A sequence a1, . . . , an of elements in a ring R is said to be left unimodular if Ra1 + ··· + Ran = R [75].

60 and both are ideals of R. 1 0 (2). R is not abelian because the idempotent is not central. 0 0 (3). This follows using (2) since the classes of left duo rings and weakly left duo rings are abelian by [72] and [115], respectively.

On the other hand, not all rings are left quasi-duo.

DD Example 4.3.5. Consider the ring R = over a division ring D. Then, R is not left DD quasi-duo.

D 0 Proof. This follows because M = is a maximal left ideal of R which is not an ideal. D 0

In the next result, we collect some useful facts for the left quasi-duo rings from [116] and [75]:

Proposition 4.3.6. For any ring R, the following statements hold:

1. If R is left quasi-duo, then so is R/A for any ideal A of R.

2. If A ⊆ J(R) is an ideal and R/A is left quasi-duo, then so is R.

3. R is left quasi-duo if and only if R/J(R) is left quasi-duo.

4.Π i∈I Ri is left quasi-duo if and only if Ri is left quasi-duo for each i ∈ I.

Proof. (1). Using the Correspondence Theorem, the maximal left ideals of the ring R/A are of the max form M/A where A ⊆ M ⊆ RR. Now, if R is left quasi-duo, then M is a right ideal of R, and so M/A is a right ideal of R/A. Thus, R/A is a left quasi-duo ring, as desired. max max (2). If R/A is left quasi-duo and M ⊆ RR, then A ⊆ J(R) ⊆ M. Thus, M/A ⊆ R/AR/A using the Correspondence Theorem, and hence M/A is an ideal of R/A by the assumption. Thus, we have M is a right ideal of R which implies that R is a left quasi-duo ring, as required. (3). This is clear using (1) and (2) above.

(4). If Πi∈I Ri is left quasi-duo, then so is Ri for each i using (1) above. The converse follows easily using the element-wise characterization for the left quasi-duo rings in Lemma 4.3.3.

A ring R is called left primitive if it has a simple faithful left R-module, and an ideal P of a ring R is called left primitive if R/P is a left primitive ring.9 Note that every division ring is simple and simple rings are left primitive, but not conversely. Below, using Corollary 4.2.4, we give a compact new proof that combines Proposition 4.1 in [75] and Proposition 1 in [58] as follows:

9For a ring R, a left R-module M is said to be faithful if l(M) = 0.

61 Proposition 4.3.7. The following statements are equivalent for a ring R [58]:

1. R is left quasi-duo

2. Every left primitive factor ring R/P is a division ring.

In particular, a left quasi-duo ring R is a division ring if and only if R is simple if and only if R is left primitive.

Proof. (1) ⇒ (2). Let S =: R/P be a left primitive ring, and let Sm be the simple faithful left S-module. As R is left quasi-duo, then S is left quasi-duo by Proposition 4.3.6. Thus, l(Sm) = l(m) is a left-max ideal of S using Corollary 4.2.4, so l(m) = 0 because Sm is faithful. That is, S is a division ring, as required. max (2) ⇒ (1). If A ⊆ RR, then P = l(R/A) = {b ∈ R | bR ⊆ A} is a left primitive ideal because max R/A is simple. Hence, R/P is a division ring by (2), and so P ⊆ RR. But, we have P ⊆ A, which implies A = P C R, proving (1). Now, every division ring is simple and simple rings are left primitive. In addition, if R is left quasi-duo and left primitive, then R is a division ring follows with P = 0 in (2), proving the last statement.

Recall that a ring R is called left noetherian if, whenever we have L1 ⊆ L2 ⊆ · · · , where each

Li is a left ideal of R, then Ln = Ln+1 for some n ∈ N. Using Proposition 4.3.7, we verify the following two results:

Example 4.3.8. For any ﬁeld F , consider the Weyl algebra W (F ) = F [x, y], where x and y are indeterminants over F and xy − yx = 1. Then, W (F ) is a simple noetherian domain [80]. On the other hand, W (F ) is neither left nor right quasi-duo by Proposition 4.3.7 because it is not a division ring.

max Proposition 4.3.9. If R is a left quasi-duo ring and A ⊆ RR, then R/A is a division ring. In particular, this is the case if A is a left-max ideal of any ring R.

Proof. Write K = R/A regarded as a simple left R-module. Then, K becomes a simple faithful left R/A-module by (r + A)x = rx for all r ∈ R and x ∈ K. Hence, the ring R/A is left primitive. On the other hand, R/A is left quasi-duo by Proposition 4.3.6. Therefore, R is a division ring by Proposition 4.3.7, as required. The last statement is clear.

62 If Ai C R for each i ∈ I and ∩i∈I Ai = 0, then the map deﬁned by r 7→ hr + Aii ∈ Πi∈I R/Ai is a one-to-one ring morphism, and we say in this case that R is a subdirect product of its images

R/Ai. A theorem of Andrunakievi˘c and Rjabuhin [3] shows that every reduced ring is a subdirect product of domains. For the left quasi-duo rings, we have the following result due to [116]:

Proposition 4.3.10. If R is a left quasi-duo ring, then R/J(R) is a subdirect product of division rings.

Proof. Let {Ai | i ∈ I} be the maximal left ideals of R. Thus, we have Ai C R, and R/Ai is a division ring for each i using Proposition 4.3.9 because R is left quasi-duo. In addition, the mapping

R → Πi∈I R/Ai given by r 7→ hr + Aii is a ring morphism with kernel J(R), and hence R/J(R) is a subdirect product of the division rings

{R/Ai}, as required.

Observe that:

Remark 4.3.11. The converse of Theorem 4.3.10 is not true in general. That is, if R/J(R) is a subdirect product of division rings, then R need not be a left quasi-duo ring in general. Indeed, the ring R in Example 5.2 [75] is neither left nor right quasi-duo although R/J(R) is a subdirect product of division rings.

So, we ask the following interesting question:

Question 4.3.12. When is a subdirect product of division rings left (right) quasi-duo?

Left Quasi-duo is not a Morita invariant Property for Rings

As we have seen in Example 4.3.5, the 2 by 2 matrix ring R over a division ring D is not left quasi-duo. It follows that:

Proposition 4.3.13. The left quasi-duo notion is not a Morita invariant property for rings.

However, we do have the following result appears in [58]. Because of its importance, we give a new shorter proof of it as follows:

63 Theorem 4.3.14. If R is left quasi-duo, then so is eRe for any e ∈ D(R).

max Proof. Write S = eRe and left X ⊆ SS. Then, RX ⊆ Re, and RX 6= Re because RX = Re max implies that S = eRe = eRX = SX = X, a contradiction. So, choose RX ⊆RM⊆ Re. Then, X = SX = eRX ⊆ eM ⊂ eRe because e∈ / M. Since eM is a left ideal of eRe, it follows that max X = eM. Now, observe that M¯ = M ⊕ R(1 − e) ⊆ RR because

R Re ⊕ R(1 − e) Re = =∼ M¯ M ⊕ R(1 − e) M

Hence, M¯ is an ideal by hypotheses. Since, MS¯ = MS, we obtain

XS = (eM)S = eMSe = eMSe¯ ⊆ eMe¯ = eM = X.

This shows that X is a right ideal of S, as required.

4.4 A “Very Semisimple-Modules” Characterization of Left Quasi- duo Rings

Following Nicholson [90], a module RM is called very semisimple (VSS) if Rm is simple for all 0 6= m ∈ M . As the name indicates, these modules are all semisimple, but not conversely. Note that 0 is very semisimple, as is every simple left module.

We ﬁrst prove the following useful result:

Lemma 4.4.1. If RM is VSS, but not simple, then l(k) = l(m) for any two non-zero elements k and m in M. This fails if M is simple.

Proof. Since RM is VSS, then Rk and Rm are both simple. So, we are in one of the following two cases: Case 1. Rk 6= Rm. In this case, the sum Rk ⊕ Rm is direct, and so l(k + m) ⊆ l(k) ∩ l(m).

But R(k + m) is also simple because RM is VSS. Thus, l(k + m) is maximal in RR, and hence we have l(k) = l(k + m) = l(m).

Case 2. Rk = Rm. As RM is not simple by hypothesis, choose x ∈ M\Rk. Then, we have Rx 6= Rk, and Rx is simple. Thus, l(x) = l(k) by Case 1. Similarly, we have l(x) = l(m), and hence l(k) = l(m), as required.

64 D 0 Now, consider the ring R = M2(D) over a division ring D. Then, M = is a simple left D 0 ideal of R. But,

1 0 0 D 0 0 D 0 l = and l = , 0 0 0 D 1 0 D 0 proving the last statement.

We also need the following fact:

∼ Lemma 4.4.2. Let A C R and B C R. If R(R/A) = R(R/B) as left R-modules, then A = B.

Proof. Let ϕ : R(R/A) → R(R/B) be an isomorphism. Write (1 + A)ϕ = r + B, for some r ∈ R.

If b ∈ B, then (b + A)ϕ = [b(1 + A)]ϕ = br + B = 0 because B C R. As ϕ is monic we have b ∈ A, and hence B ⊆ A. Similarly, we have A ⊆ B.

We are now ready to give a compact proof of the following main result, due to [90], which gives a module-theoretic characterization of the left quasi-duo rings in terms of their VSS modules as follows:

Theorem 4.4.3. The following statements are equivalent for a ring R:

1. R is left quasi-duo.

2. Every homogeneous, semisimple left R-module is very semisimple.

3. K ⊕ N is very semisimple for any simple left R-modules K and N with K =∼ N.

Proof. (1) ⇒ (2). Let 0 6= k ∈ M; we must prove that Rk is simple. Since k ∈ M, we have ∼ k ∈ Rx1 ⊕ Rx2 ⊕ · · · ⊕ Rxm where each Rxi is simple and Rxi = Rxj for all i, j because M is homogenous and semisimple. If we choose m minimally, then k = k1 + k2 + ··· + km where m 0 6= ki ∈ Rxi for each i. Hence, Rki = Rxi is simple for each i. Because ⊕i=1Rki is direct, we obtain l(k) = l(k1) ∩ l(k2) ∩ · · · ∩ l(km). But l(k1) = l(k2) = ··· = l(km) follows using Lemma 4.4.2 ∼ ∼ ∼ because R/l(ki) = Rki = Rkj = R/l(kj) where l(kj) and l(ki) are left-max ideals of R for all i, j because R is a left quasi-duo ring by (1). Hence, l(k) = l(k1) is a maximal left ideal of R which implies that Rk is simple, proving (2). (2) ⇒ (3). This is obvious.

(3) ⇒ (1). Let L be a maximal left ideal of R; we must show that L C R, that is Lr ⊆ L.

Consider the module RX = R/L ⊕ R/L and let x = (1 + L, r + L) ∈ X. To show that Lr ⊆ L,

65 it suﬃces to show that Lx = 0. Suppose on the contrary that Lx 6= 0. But X is homogenous and semisimple, so (3) shows that Rx is simple. It follows that Lx = Rx, whence x = tx for some t ∈ L. This means that x = (1 + L, r + L) = (t + L, tr + L), from which 1 + L = L. This is the desired contradiction.

Homogenous semisimple left R-modules over a left quasi-duo ring R enjoy the following nice properties:

Corollary 4.4.4. Let R be a left quasi-duo ring. If M is a nonzero homogenous semisimple left

R-module and A = l(M) C R, then the following statements hold:

max 1. A = l(m) for any 0 6= m ∈ M and A ⊆ RR.

∼ (I) 2. D =: R/A is a division ring, M = D for some set I, and end(RM) = end(DM) where M is a left D-module via (r + A)m = rm for all r ∈ R and m ∈ M.

Proof. Assume that RM is a nonzero homogenous semisimple left module, then it is VSS using Theorem 4.4.3 because R is left quasi-duo. Now, if M is simple, then the result follows using Proposition 4.2.3. So, assume that M is not simple, then in this case we have A = l(m) for any 0 6= m ∈ M by Lemma 4.4.1. Now, since Rm is simple for any 0 6= m ∈ M being VSS, it follows that D =: R/A = R/l(m) is simple as a left R-module, and so is a division ring using Proposition 4.2.3 because l(m) is a left-max ideal. Moreover, M =DM via (r +A)m = rm, so end(RM) = end(DM). ∼ (I) Finally, since RM is homogenous and semisimple, it follows that M = K for some set I, where K = Rm =∼ R/l(m) = R/A = D and m is any non-zero element in M, as required.

4.5 Generalized Upper Triangular Matrix Ring over Left Quasi- duo Rings

As mentioned earlier, the left quasi-duo notion is not a Morita invariant property for rings because it does not pass to the matrix rings as we have seen in Example 4.3.5. On the other hand, the 2 × 2 upper triangular matrix ring over a division ring D in Example 4.3.4 is left quasi-duo. More generally, a ring R is left quasi-duo if and only if the upper triangular matrix ring over R is left quasi-duo using Proposition 2.1 in [116]. In this section, we extend this result to any n × n generalized upper triangular matrix ring.

66 For the n × n generalized upper triangular matrix ring, we prove the following result which will be one of our keys to prove, in a later section, one of the main results in the thesis that characterises the left quasi-duo rings which are I-ﬁnite:

Theorem 4.5.1. Let R1,...,Rn be rings, and let R = Tn[Ri,Vij] be a generalized upper triangular matrix ring over Ri. Then:

∼ 1. J(R) = Tn[J(Ri),Vij] and R/J(R) = Πi[Ri/J(Ri)].

2. R is left quasi-duo if and only if each Ri is left quasi-duo.

Proof. (1). Deﬁne the map

ϕ : R → Πi[Ri/J(Ri)] by ϕ[ri, vij] = hri + J(Ri)i.

Then, ϕ is an onto ring morphism with kernel Tn[J(Ri),Vij]. Since J(Πi[Ri/J(Ri)]) = 0 it follows that J(R) ⊆ ker(ϕ) = Tn[J(Ri),Vij]. On the other hand, a routine computation shows that

Tn[J(Ri),Vij] is a quasi-regular ideal of R which implies that Tn[J(Ri),Vij] ⊆ J(R). Hence, we ∼ have Tn[J(Ri),Vij] = J(R), and so R/J(R) = Πi[Ri/J(Ri)], as required.

(2). If R is left quasi-duo, then so is its corner eiiReii for each i by Theorem 4.3.14. Hence, ∼ each Ri is left quasi-duo because we have Ri = eiiReii for each i, where eii is the matrix unit.

Conversely, if each Ri is left quasi-duo, then so also is each Ri/J(Ri) by (3) of Proposition 4.3.6, ∼ and hence Πi[Ri/J(Ri)] = R/J(R) is left quasi-duo using (1) above and (4) of Proposition 4.3.6. Therefore, R is left quasi-duo by (3) of Proposition 4.3.6, as required.

Using Theorem 4.5.1, we immediately get:

Corollary 4.5.2. The split-null extension   RV Λ =   0 S is left quasi-duo if and only if both S and R are left quasi-duo [65].

We also have:

Corollary 4.5.3. A ring R is left quasi-duo if and only if the n × n upper triangular matrix ring over R is left quasi-duo [116].

67 Chapter 5

Advanced Results for Left Quasi-duo Rings

In this chapter, we continue our study for the class of left quasi-duo rings. In particular, we prove three of the major results in the thesis.

5.1 Left Quasi-duo Rings with Finite Left Width

In this section, we define a new notion for rings called the left width. In addition, we prove one of our major results; the Width Theorem, where we then use it to get some new facts about the left quasi-duo rings in terms of the width notion. We also give a characterization of any left quasi-duo ring that has a finite number of non-isomorphic simple left modules (that is, with finite left width). This in particular shows that the “left quasi-duo” notion coincides with the “right quasi-duo” notion which gives a partial answer to the open question; whether or not the “quasi-duo” notion is a left-right symmetric property for rings.

5.1.1 The Width Theorem and the Left Width of Rings

We need the following preparation before we get ready to prove the Width Theorem.

First, we introduce the following notion:

Deﬁnition 5.1.1. For a ring R, call a left R-module K ideal-simple if it is simple and l(K) = l(k) is an ideal of R whenever 0 6= k ∈ K.

68 Next, we prove the following key result:

Lemma 5.1.2. For a ring R, the following statements are true:

1. Let RK be a left module over R. Then, l(K) is left-max if and only if RK is ideal-simple.

2. Let A be a left ideal of R. Then, A is left-max if and only if R(R/A) is ideal-simple.

Proof. (1). Assume ﬁrst that l(K) is a left-max ideal of R. Now, given 0 6= k ∈ K, we have l(K) ⊆ l(k) 6= R, and so l(K) = l(k) by maximality of l(K). Hence, we are done because l(K) C R.

Conversely, let RK be ideal-simple, and choose 0 6= k ∈ K such that l(K) = l(k) C R. Then, R/l(K) = R/l(k) =∼ Rk = K is simple, and so l(K) is a maximal left ideal of R. Hence, l(K) is left-max in R, as required.

(2). This follows using (1) with RK=R(R/A) and l(K) = A.

We notice that:

Remark 5.1.3. For a simple ring R, R has a left-max ideal if and only if R is a division ring.

Thus, we have:

Example 5.1.4. The Weyl algebra W (R) has no left-max ideal because it is a simple ring that is not a division ring. Notice also that the ring W (R) is not left quasi-duo.

We ﬁx the following notation for any ring R:

(i) Deﬁne

F = F(R) = {A | A is left-max in R} (possibly empty). (5.1)

(ii) For any left R-module M, write

∼ class(M) = {RN | RN = RM}. (5.2)

(iii) Let

{Ki | i ∈ I} (5.3) be a system of distinct representatives (SDR) of the isomorphism classes of ideal-simple left R-modules.

69 (iv) Finally, deﬁne

C = C(R) = {class(Ki) | i ∈ I}. (5.4)

We are now in a position to prove the major result in this section:

Theorem 5.1.5 (Width Theorem). For any ring R, deﬁne F, C and {Ki | i ∈ I} as above. Then, we have:

1. |F| = |C| = |I|.

0 2. F = {l(Ki) | i ∈ I}. Hence, C = {R/A | A ∈ F} is an SDR of the isomorphism classes of ideal-simple left R-modules.

3. If l(Ki) = l(Kj), then i = j.

Proof. We have two cases: Case (1): R has no left-max ideals, then R has no ideal-simple left modules by Lemma 5.1.2. So, we are done in this case because we have |F| = |C| = |I| = 0. Case (2): R has at least one left-max ideal, equivalently, R has at least one ideal-simple left module. Then, it follows that the sets F, C and I are non-empty, and so |F| 6= 0, |C| 6= 0 and |I|= 6 0. So, we prove the three statements as follows:

(1). First, deﬁne the map

σ : I → C by σ(i) = class(Ki).

∼ If class(Ki) = class(Kj), then Ki = Kj, and so i = j by the deﬁnition of the Ki in (5.3). In addition, σ is clearly onto, and hence σ is a bijection, proving |C| = |I|. Now, we deﬁne the following two functions:

Φ: F → C by Φ(A) = class(R/A) for all A ∈ F.

Ψ: C → F by Φ(class(K)) = l(K) for any ideal-simple RK.

First, notice that for any A ∈ F, we have class(R/A) ∈ C because R/A is ideal-simple using

Lemma 5.1.2, and similarly l(K) ∈ F for any ideal-simple RK. We next check that Ψ is well deﬁned. To this end, suppose that class(K) = class(N) where RK and RN are ideal-simple. Choose 0 6= k ∈ K and 0 6= n ∈ N, and write A = l(K) = l(k) and B = l(N) = l(n). Then, as

70 mentioned above A, B ∈ F, and we also have R/A =∼ K =∼ N =∼ R/B. Thus, A = B using Lemma 4.4.2, and hence l(K) = l(N), as required.

To see that |F| = |C|; we show that Φ and Ψ are mutually inverse. For Ψ ◦ Φ = 1F ; if A ∈ F, then we have:

(Ψ ◦ Φ)(A) = Ψ(Φ(A)) = Ψ(class(R/A)) = l(R/A) = A = 1F (A).

Now, let RK be ideal-simple and choose 0 6= k ∈ K such that l(K) = l(k) C R. Then, we have:

(Φ ◦ Ψ)(class(K)) = Φ(Ψ(class(K))) = Φ(l(K)) = class(R/l(K)) = class(R/l(k)), but

class(R/l(k)) = class(Rk) = class(K) = 1C(class(K)).

Hence, Φ ◦ Ψ = 1C. Therefore, we have |F| = |C| = |I|, proving (1).

(2). Assume ﬁrst that A is a left-max ideal, then A = l(K) where RK =RR/A is an ideal-simple ∼ using Lemma 5.1.2. By (5.3), we have K = Ki for some i ∈ I, and so A = l(K) = l(Ki). Now, if

Ki is an ideal-simple, then l(Ki) is a left-max ideal again using Lemma 5.1.2 which implies that l(Ki) = A for some A ∈ F, as required. The last statement is clear.

(3). Suppose that l(Ki) = l(Kj). Then, l(Ki) = l(Kj) ∈ F by (2). Hence,

Φ(l(Ki)) = Φ(l(Kj)), and so class(Ki) = class(Kj).

Thus, i = j because σ is one-to-one in the proof of (1), as required.

Using the Width Theorem, we deﬁne the notion of left width for rings as follows:

Deﬁnition 5.1.6. Let R be any ring. The left width of R, denoted by wl(R), is deﬁned by

wl(R) = |F| = |C| = |I|

Thus, wl(R) = 0 if and only if R has no left-max ideals if and only if R has no ideal-simple left modules.

In the next example, we compute the left width for some rings as follows:

Example 5.1.7. 1. If R is a local ring, then wl(R) = 1.

2. For the Weyl algebra W (R), we have wl[W (R)] = 0.

3. If R = R × W (R), then wl(R) = 1.

71 4. wl(D) = 1 for any division ring D.

5. wl(R) ≥ 1 for any left (right) quasi-duo ring R. In particular, this is the case for any commutative ring R.

Proof. (1). J(R) is the unique left-max ideal in any local ring. Hence, wl(R) = 1. (2). This follows because W (R) has no left-max ideals by Example 5.1.4.

(3). 0 × W (R) is the unique left-max ideal in R, and so wl(R) = 1. (4). This follows since the zero ideal is the only left-max ideal in any division ring D. (5). Any ring R has at least one maximal left ideal L using Zorn’s Lemma. On the other hand, L is a left-max ideal because R is a left quasi-duo ring, and so we have wl(R) ≥ 1. The last statement follows since commutative rings are left (right) quasi-duo.

Three Useful Results

We end this part proving the following three useful results that will be used later.

Proposition 5.1.8. Let R be a ring. If B ⊆ J(R), then wl(R/B) = wl(R).

Proof. Write F(R) = {A | A is left-max in R} as in (5.1). Then, using the Correspondence Theorem, it follows that F(R/B) = {A/B | A ∈ F(R)}. Now, deﬁne the map:

ϕ : F(R) → F(R/B) by ϕ(A) = A/B.

Thus, the result follows because ϕ is one-to-one and onto.

Proposition 5.1.9. If R is any ring and n ≥ 2, then wl[Mn(R)] = 0.

Proof. If X ⊆ Mn(R) is a left-max ideal, then X C Mn(R), and so X = Mn(A) for some A C R. ∼ Hence, Mn(R/A) = Mn(R)/Mn(A) is a division ring by Proposition 4.3.9, a contradiction since by our assumption n ≥ 2.

Turning to the direct products. If R = Πi∈I Ri is a direct product of rings, and A ⊆ Rk is a left ideal of Rk, write A¯k = Πi∈I Xi where Xi = Ri if i 6= k, and Xk = A. Then, we have:

A ∈ F(Rk) if and only if A¯k ∈ F(R).

We call these A¯k the ordinary left-max ideals of R. Note that A¯k = A¯m if and only if k = m.

Hence, given k ∈ I, the set Ak = {A¯k | A ∈ F(Rk)} consists of left-max ideals of R. Thus, the set of all ordinary left-max ideals in R is the disjoint union ∪· k∈I Ak. So, we obtain:

72 Proposition 5.1.10. Let R = Πi∈I Ri where each Ri is a ring. Then:

1. wl(R) ≥ Σi∈I wl(Ri).

2. wl(R) = Σi∈I wl(Ri) if I = {1, 2, 3, . . . , n}.

Proof. (1). This follows by the discussion above. (2). In this case every left-max ideal of R is ordinary, and so we have equality in (1).

5.1.2 The Width Theorem for Left Quasi-duo Rings

We now use Theorem 5.1.5 to get the following result for the class of left quasi-duo rings, where part (2) gives us a characterization of any left quasi-duo ring R with ﬁnite left width:

Theorem 5.1.11 (Width Theorem for Left Quasi-duo Rings). Let R be a ring with left width wl(R). Let {Ki | i ∈ I}, F = {A | A is left-max in R} and C = {class(Ki) | i ∈ I} as in (5.1),

(5.2), (5.3) and (5.4). For each i ∈ I, write Ai = l(Ki). Then, we have:

1. If R is left quasi-duo, then the following statements hold:

max (a) F = {Ai | i ∈ I} = {A | A ⊆ RR}. Moreover, if Ai = Aj, then i = j.

0 (b) The set C = {R/Ai | i ∈ I} is an SDR of the isomorphism classes of the simple left R-modules.

2. The following two statements are equivalent for R:

(a) R is left quasi-duo and wl(R) = n is ﬁnite.

(b) R/J(R) is a ﬁnite direct product of division rings.

Proof. Part (1). Notice ﬁrst that the ideal-simple left modules for any left quasi-duo ring R are the simple left modules over R using Corollary 4.2.4, and so {Ki | i ∈ I} is an SDR of the isomorphism classes of the simple left R-modules. Moreover, any left quasi-duo ring R has at least one left-max ideal using Example 5.1.7, and hence the two sets F and C are non-empty. max (a). Using (2) of Theorem 5.1.5, we have F = {Ai | i ∈ I}. In addition, F = {A | A ⊆ RR} because R is a left quasi-duo ring. The last statement follows by (3) of Theorem 5.1.5, which ends the proof of (a).

73 ∼ (b). For each i ∈ I, we have R/Ai = R/l(Ki) = R/l(ki) = Rki = Ki using Corollary 4.2.4, proving (b). Part (2). (a) ⇒ (b). Assume that R is a left quasi-duo ring. Since by assumption we have n wl(R) = n < ∞, let A1,A2, ··· ,An be all the left-max ideals of R. Then, we have J(R) = ∩i=1Ai.

On the other hand, Ai + Aj = R whenever i 6= j, and each Ai C R being a left-max ideal of R. ∼ n So, we have R/J(R) = Πi=1R/Ai using the Chinese Remainder Theorem. Finally, each R/Ai is a division ring using Proposition 4.3.9, proving (b). ∼ n (b) ⇒ (a). Assume now that R/J(R) = Πi=1R/Ai is a product of n division rings. Hence, R/J(R) is left quasi-duo using Proposition 4.3.6, which implies that R is left quasi-duo again using

Proposition 4.3.6. In addition, for each division ring R/Ai, we have wl(R/Ai) = 1 using (4) of n Example 5.1.7. Hence, we have wl(R/J) = Σi=1wl(R/Ai) = n by Proposition 5.1.10. Therefore, wl(R) = n using Proposition 5.1.8, proving (a).

Recall that a ring R is called I-ﬁnite if it has no inﬁnite sets of orthogonal idempotents. Now, using Theorem 5.1.11, we get the following two important consequences:

It is still an open question since Yu [116] introduced these rings in 1993, whether or not the “left quasi-duo” notion coincides with the “right quasi-duo” notion. The following consequence has a partial answer to this question:1

Corollary 5.1.12. Let R be any ring with wl(R) = n < ∞. Then, the following two statements are equivalent:

1. R is left quasi-duo.

2. R is right quasi-duo.

∼ n Proof. (1) ⇒ (2). Assume that R is left quasi-duo. Then, R/J(R) = Πi=1R/Ai is a product of n division rings using (2) of Theorem 5.1.11 because wl(R) = n < ∞ by assumption. It follows that R is right quasi-duo using the right version of Proposition 4.3.6 and the fact that division rings are right quasi-duo. (2) ⇒ (1). Similar to the above argument.

1Notice that this has been brieﬂy discussed after Question 5.1 in [75]. Here, we formally prove the statement using the new notion of left width.

74 Hence, the following question is still of interest:

Question 5.1.13. Is the “quasi-duo” notion a left-right symmetric property for rings?

Local rings are left quasi-duo as discussed before Example 4.3.4. In addition, we have wl(R) = 1 for any local ring R using Example 5.1.7. Hence, we get:

Corollary 5.1.14. A ring R is local if and only if R is left quasi-duo and wl(R) = 1.

Proof. If R is a local ring, then R is left quasi-duo and wl(R) = 1 using the discussion above.

Conversely, assume that R is left quasi-duo and wl(R) = 1. Then, using (2) of Theorem 5.1.11, we ∼ n have R/J(R) = Πi=1R/Ai is a product of n division rings. Now, since wl(R) = 1, it follows that n = 1, and so R/J(R) is a division ring, that is, R is a local ring, as required.

Notice that the left quasi-duo assumption in Corollary 5.1.14 is important and can not be dropped. Indeed, the ring R = R × W (R) in Example 5.1.7 is neither left quasi-duo nor local, but wl(R) = 1.

In the next Example, we compute the left width for some left quasi-duo rings as follows:

a Example 5.1.15. 1. If R = Z(p,q) = { b ∈ Q | p - b, q - b}, then wl(R) = 2.

n 2. If R = D where D is a division ring and n ≥ 1, then wl(R) = n.

DD 3. For a division ring D, let R = . Then, wl(R) = 2. 0 D

4. wl(Z) = ∞.

th 5. If pi ∈ Z is the i prime and R = Πi∈I Zpi . Then:

(a) wl(R) = ∞ if |I| = ∞.

(b) wl(R) = n if |I| = n < ∞.

∼ Proof. (1). R/J(R) = Zp × Zq is a left quasi-duo ring with exactly two left-max ideals. Hence, we have wl[R/J(R)] = 2, and so wl(R) = 2 using Proposition 5.1.8. (2). We prove the result for n = 2. R in this case has only the following two left-max ideals:

A1 = D × 0 and A2 = 0 × D.

Hence, wl(R) = 2, as required. (3). This follows because R has only the following two left-max ideals:

75 DD 0 D M = and N = . 0 0 0 D

(4). The left-max ideals in Z are of the form pZ, where p is a prime number in Z. Hence, wl(Z) = |P| which is inﬁnite, where P is the set of all prime numbers in Z.

(5). For any prime pi ∈ Z, we have wl(Zpi ) = 1 using (4) of Example 5.1.7 because Zpi is a division ring. Hence, the result follows using Proposition 5.1.10.

Observe that the left quasi-duo ring R in (1) of Example 5.1.15 has ﬁnite left width, but it is not semiperfect because idempotents do not lift module J(R). So, the following question is of interest:

Question 5.1.16. Let R be a left quasi-duo ring with wl(R) = n < ∞. When is R semiperfect?

Finally, notice that using (2) of Theorem 5.1.11 any left quasi-duo ring with ﬁnite left width is

I-finite. On the other hand, the converse is not true in general. Indeed, the ring of integers Z is left quasi-duo and I-finite, but it has infinite left width as we have seen in (4) of Example 5.1.15. So, we turn our attention to the class of I-finite left quasi-duo rings and give a characterization of them in the next section.

5.2 Left Quasi-duo Rings that are I-ﬁnite

In this section, we study the left quasi-duo rings that are I-finite. We first give a characterization of the left quasi-duo I-finite rings that are semiprime. Next, we prove a useful fact for any left quasi-duo ring R that is not a division ring where its left socle is not contained in its Jacobson radical. This fact will be the key to proving one of the main results in this thesis which characterises the class of left quasi-duo rings that are I-finite.

5.2.1 Left Quasi-duo I-ﬁnite Rings that are Semiprime

As mentioned earlier, a ring R is called I-ﬁnite if it has no inﬁnite set of orthogonal idempotents, and called semiprime if it has no nonzero nilpotent one-sided (two-sided) ideals.

First, in the next result we give a nice characterization of the left quasi-duo I-ﬁnite rings that are semiprime as follows:

Theorem 5.2.1. The following statements are equivalent for any ring R:

1. R is left quasi-duo, I-ﬁnite and semiprime.

76 ∼ 2. R = D1 × D2 × · · · × Dn × S where each Di is a division ring, and S is either zero or it is left quasi-duo, semiprime, I-ﬁnite and r(M) ⊆ M for every left-max ideal M of S.

Proof. (1) ⇒ (2). If R is the zero ring, then the statement is trivially true. So, assume that R is a non-zero ring. Since R is left quasi-duo, let A be a left-max ideal of R. As we have

[A ∩ r(A)]2 ⊆ r(A)A = 0, it follows that A ∩ r(A) = 0 because R is semiprime. Now, if r(A) ⊆ A for every left-max ideal of R, then we are done. So, assume that r(A) * A, then it follows that ∼ R = r(A) ⊕ A = D1 × R1

∼ ∼ where D1 = r(A) is a division ring by Proposition 4.3.9, and R1 = A is left quasi-duo, semiprime and I-finite. If R1 = 0, then we are done. So, assume that R1 6= 0, and let A1 be a left-max ideal in ∼ R1 with r(A1) * A1. Then, as before we obtain R = D1 ×D2 ×R2 where D2 is a division ring. But, since R is I-finite by assumption, this process cannot continue indefinitely and the result follows. (2) ⇒ (1). This is clear because division rings enjoy the three ring properties in (1).

Our next goal is to give a non-semiprime version of Theorem 5.2.1, and give a characterization of the class of I-finite left quasi-duo rings. To facilitate the discussion, we first recall the notion of a frame of any ring R, and use it to define a special class of the generalized upper triangular matrix rings introduced in Section 4.5.

Let R be a ring, and let n be a positive integer number. A frame of R with length n, denoted by {ei; n}, is a decomposition 1 = e1 + e2 + ··· + en, where {e1, e2, . . . , en} is a set of orthogonal 2 idempotents in R. A ring R is semiperfect if and only if it has a frame {ei; n} for some n ≥ 1 where each corner eiRei is a local ring. More generally, any I-ﬁnite ring R has a frame {ei; n} for some n ≥ 1 where each ei is a primitive idempotent, that is, each corner ring eiRei contains no idempotent except 0 and ei.

A frame {ei; n} of a ring R will be called an upper triangular frame if eiRej = 0 whenever i > j. The reason for the name is that, in this case, the Pierce decomposition3 of R has the following upper triangular form

2 {e1, e2, . . . , en} is a set of orthogonal idempotents in R if eiej = 0 for any i 6= j. 3Let e and f be any two idempotents in a ring R. Then, the decomposition R = eRe ⊕ eRf ⊕ fRe ⊕ fRf is called eRe eRf ere erf the Pierce decomposition [103] of R. In this case, we have R ∼ via r 7→ . = fRe fRf fre frf

77   e1Re1 e1Re2 . . . e1Ren−1 e1Ren      0 e2Re2 . . . e2Ren−1 e2Ren    ∼  ......  R =  . . . . .       0 0 . . . en−1Ren−1 en−1Ren   0 0 ... 0 enRen

Remark 5.2.2. If each eiRei has an upper triangular frame of length mi. Then, the union of the idempotents in these frames is again an upper triangular frame of R with length m1 + m2 + ··· mn.

For convenience, we write Ri = eiRei and Vi,j = eiRej when i 6= j. Then, the above matrix ring of R becomes   R1 V1,2 ...V1,n−1 V1,n      0 R2 ...V2,n−1 V2,n     ......   . . . . .       0 0 ...Rn−1 Vn−1,n   0 0 ... 0 Rn which is just a generalized upper triangular matrix ring Tn[Ri,Vij] over the rings Ri = eiRei.

Next, we recall that the left socle of a ring R is given by:

P Sl(R) = S(RR) = {K ⊆ R | K is a simple left ideal}, with a similar notation for the right socle. And we set Sl(R) = 0 if R has no simple left ideals.

5.2.2 Characterization of Left Quasi-duo I-ﬁnite Rings

The left quasi-duo rings have a close connection to upper triangular matrix rings. The following result is the key to understanding this:

Proposition 5.2.3. Let R be a left quasi-duo ring that is not a division ring. If Sl(R) * J(R), then R has an upper triangular frame {e1, e2} such that:

1. e1Re1 is a nonzero division ring.

2. e2Re2 is a nonzero left quasi-duo ring.

Hence, R is an upper triangular matrix ring with a frame of length at least 2.

78 Proof. Since we have Sl(R) * J(R), then there exists a simple left ideal K of R such that K * J(R). Hence, by Brauer’s Lemma we have:

K = Re for some nonzero idempotent e ∈ R.

∼ So, R(1 − e) is a maximal left ideal of R because R/R(1 − e) = Re is simple. Hence, R(1 − e) C R because R is left quasi-duo by assumption. In particular, we have (1 − e)R ⊆ R(1 − e), which implies that (1 − e)Re = 0. Hence, the Pierce decomposition of R is   eRe eR(1 − e) R =∼   0 (1 − e)R(1 − e)

∼ Finally, as Re is simple, we have eRe = endR(Re) is a division ring using Schur’s Lemma, and (1−e)R(1−e) is a left quasi-duo ring using Theorem 4.3.14. In addition, we have (1−e)R(1−e) 6= 0 since 1 6= e as R(1 − e) is a maximal left ideal. If we take e1 = e and e2 = 1 − e, then {e1, e2} is a frame of R with length 2. Therefore, R is an upper triangular matrix ring with a frame of length at least 2, as desired.

Having done the preparation steps, we are now in a position to prove one of the major results in the thesis which gives a nice characterization of the I-ﬁnite left quasi-duo rings as follows:

Theorem 5.2.4. The following two conditions are equivalent for a ring R:

1. R is left quasi-duo and I-ﬁnite.

∼ 2. There exists an integer n ≥ 1 such that R = Tn[Ri,Vij] is a generalized upper triangular

matrix ring where Ri 6= 0 for each i, and either (a) or (b) holds:

(a) Ri is a division ring for each i. Hence, R is a semiperfect ring.

(b) Ri is a division ring for each i < n, and Rn is left quasi-duo, I-ﬁnite, but not a division

ring with Sl(Rn) ⊆ J(Rn).

Proof. (1) ⇒ (2). The result holds if R is the zero ring, so assume that R is a nonzero ring. If R is a division ring, then (a) holds with n = 1 and R1 = R. Also, if R is not a division ring with

Sl(R) ⊆ J(R), then (b) holds with n = 1 and R1 = R. So, assume that R is not a division ring and

Sl(R) * J(R). Hence, by Proposition 5.2.3, we have   e1Re1 e1Rf1 R =∼   0 f1Rf1

79 for some idempotents e1 and f1, and hence R has an upper triangular frame {e1, f1} such that e1Re1 =: R1 is a nonzero division ring, and f1Rf1 =: S1 is a nonzero left quasi-duo ring.

Now, if S1 is a division ring, then we are in Case (a) with R2 = S1; and if S1 is not a division ring such that Sl(S1) ⊆ J(S1), then we are in Case (b) with R2 = S1. So, assume that S1 is not a division ring and Sl(S1) * J(S1), then again using Proposition 5.2.3 we have   e (S )e e (S )f ∼ 2 1 2 2 1 2 S1 =   0 f2(S1)f2

for some idempotents e2 and f2, and S1 in this case has an upper triangular frame {e2, f2} such that e2(S1)e2 =: R2 is a nonzero division ring, and f2(S1)f2 =: S2 is a nonzero left quasi-duo ring.

In addition, {e1, e2, f2} is an upper triangular frame of the ring R as mentioned in Remark 5.2.2.

Now, as before we are in Case (a) if S2 is a division ring with R3 = S2, and in Case (b) with

R3 = S2 if S2 is not a division ring such that Sl(S2) ⊆ J(S2). So, if both fail we repeat the process.

But, because the frame of R constructed above gets longer and longer, the I-ﬁnite hypothesis of R guarantees that this process cannot continue indeﬁnitely. So, at some stage, say n, we have   R1 V1,2 ...V1,k−1 V1,k      0 R2 ...V2,k−1 V2,k    ∼  ......  R =  . . . . .  = Tn[Ri,Vij]      0 0 ...Rn−1 Vk−1,k   0 0 ... 0 Rn where Ri is a division ring for each i < n, and Rn = Sn−1 is either a division ring and thus we are in Case (a), and hence R is a semiperfect ring in this case; or Rn is not a division ring with

Sl(Rn) ⊆ J(Rn), and so we are in Case (b), as desired. ∼ (2) ⇒ (1). Given (2), we have R = Tn[Ri,Vij] where each Ri is a left quasi-duo ring in both cases (a) and (b). Hence, R is left quasi-duo using Theorem 4.5.1. Finally, deﬁne the map

θ : R → R1 × · · · × Rn by θ[ri, vij] = hrii.

Then θ is an onto ring morphism with ker(θ) ⊆ J(R). Since R1 × · · · × Rn is I-finite (the Ri are I-finite by hypothesis), it follows that R is I-finite.

Notice that there is an I-ﬁnite left quasi-duo ring R with Sl(R) ⊆ J(R), but not semiperfect as we can see in the next example:

80 Example 5.2.5. If R is left quasi-duo and I-ﬁnite, then idempotents need not lift modulo J(R).

n Proof. Consider the ring R = Z(p,q) = { d | p - d and q - d} in Example 1.2.14. Then, R is left quasi-duo being commutative, I-ﬁnite, and Sl(R) ⊆ J(R) because it is an integral domain. On the other hand, idempotents do not lift modulo J(R), and hence R is not semiperfect.

Finally, in view of Theorem 5.2.4 and Example 5.2.5:

Question 5.2.6. We ask the following two interesting questions:

1. If R is left quasi-duo and I-ﬁnite with Sl(R) ⊆ J(R), when is R semiperfect?

2. What do the left quasi-duo rings R with Sl(R) ⊆ J(R) look like?

5.3 Left-Max idealtor and Left-Max stable Rings:

We conclude our work by giving a connection between the two main parts of the thesis; left L-stability in rings and left quasi-duo rings. We first define the left-max idealtor X for any ring R in term of its maximal left ideals. Our hope when we defined this left idealtor was to show that the left quasi-duo rings are afforded by such an idealtor, but unfortunately this is not the case as we will see later. Indeed, it turns out that the class of left quasi-duo rings is not affordable in general. However, we study the class of left X -stable rings which will be called the left-max stable rings, where a characterization of them will be given.

We start by proving the following fundamental lemma:

max Lemma 5.3.1. Let L ⊆ RR. Write K = R/L with r¯ = r + L for any r ∈ R, and abbreviate U = U(R), K∗ = K\{0¯}. Then, the following statements are equivalent:

1. Ra + L = R with a ∈ R, implies that a − u ∈ L for some u ∈ U.

2. If a, b ∈ R\L, then ua¯ = v¯b for some u, v ∈ U.

3. K∗ = Ua¯ for any a¯ ∈ K∗.

Proof. (1) ⇒ (2). If a ∈ R\L, then Ra + L = R because L is a maximal left ideal of R. So, using (1), let a − u−1 ∈ L with u ∈ U. Hence, ua − 1 ∈ L, which implies that ua¯ = 1.¯ Similarly, if b ∈ R\L, then v¯b = 1.¯ Therefore, ua¯ = v¯b for some u, v ∈ U, as required.

81 (2) ⇒ (3). We always have Ua¯ ⊆ K∗ for anya ¯ 6= 0.¯ Now, if ¯b ∈ K∗, (2) gives ua¯ = v¯b for some u, v ∈ U. Hence, ¯b = (v−1u)¯a ∈ Ua¯, proving (3). (3) ⇒ (1). If Ra + L = R with a ∈ R, then a∈ / L. Hence, K∗ = Ua¯ using (3). Thus, ua¯ = 1,¯ which implies that ua − 1 ∈ L. Therefore, a − u−1 ∈ L, proving (1).

Hence,

Deﬁnition 5.3.2. A maximal left ideal L of a ring R is said to be a left-max stable ideal if the conditions in Lemma 5.3.1 hold for L. In addition, the left idealtor X deﬁned by

max X (R) = {L | L ⊆ RR} for each ring R will be called the left-max idealtor. Call a ring R left-max stable if it is left X -stable, and so R is a left-max stable ring if and only if every maximal left ideal of R is left-max stable.

Example 5.3.3. The 2 by 2 matrix ring M2(D) over a division ring D is a left-max stable ring.

Proof. The ring R = M2(D) is unit-regular by [49], and hence R is left L-stable for any left idealtor L using Theorem 1.4.13. In particular, it is left X -stable, and so R is a left-max stable ring, as required.

More generally:

Example 5.3.4. Division rings, local rings, unit-regular rings and SR1 rings are all left-max stable rings.

Proof. These rings are left L-stable for any left idealtor L using Theorem 1.4.13 and Theorem 1.4.3. In particular, each of them is left X -stable, where X is the left-max idealtor, and so each is a left-max stable ring, as required.

In view of Example 5.3.4 and since the SR1 rings are left UG, IC and DF, we ask:

Question 5.3.5. Are left-max stable rings left UG, IC or DF?

On the other hand, not all rings are left-max stable as the following key example veriﬁes:

Example 5.3.6. The (left) max stable ideals of the ring of integers Z are 2Z and 3Z. Hence, Z is not a (left) max stable ring.

82 Proof. Observe ﬁrst that the maximal (left) ideals of Z are of the form pZ where p is a prime ¯ ∼ number. Now, if 0 6=a ¯ ∈ Z/pZ = Zp, then Ua¯ = {a,¯ −a¯}. Hence, pZ is a left-max stable if ∗ ¯ Zp = Ua¯ = {a,¯ −a¯}, and so Zp = {0, a,¯ −a¯}. But, |Zp| = p, so we must have p = 2 ifa ¯ = −a¯; and p = 3 ifa ¯ 6= −a¯, as required. The last statement is clear.

One may expect that the left-max idealtor aﬀords the class of left quasi-duo rings, but unfortunately this is not the case because:

Proposition 5.3.7. In general, we have:

1. Neither left quasi-duo rings nor left-max stable rings implies the other. In particular, the left-max idealtor does not aﬀord the class of left quasi-duo rings.

2. The class of left quasi-duo rings is not aﬀordable.

Proof. (1). The ring of integers Z is left quasi-duo being commutative, but it is not left-max stable as we have seen in Example 5.3.6. In addition, if R is a left-max stable ring, then it may not be left quasi-duo because the ring R in Example 5.3.3 is a left-max stable ring, but it is not left quasi-duo as we have seen in Example 4.3.5. Thus, we conclude that the left-max idealtor does not aﬀord the left quasi-duo rings. (2). The result follows using Theorem 1.4.3 because the ring R in Example 4.3.5 is SR1 but not left quasi-duo.

So, the following question is of interest:

Question 5.3.8. When is the class of left quasi-duo rings aﬀordable? When are left quasi-duo rings left-max stable?

∗ Observe that if RM is any left R-module and 0 6= m ∈ M. Then, clearly M = M ∪·{0} is a disjoint union, and we always have 0 ∈/ Um with U = U(R). Hence, Um ∪·{0} is a disjoint union and Um ⊆ M ∗. Therefore,

Um ∪·{0} ⊆ M ∗ ∪·{0} = M always holds, with equality if and only if Um = M ∗.

In the following result, we give a characterization of the left-max stable rings in terms of their simple modules as follows:

83 Theorem 5.3.9. Let R be any ring, and write U = U(R). Then, the following statements are equivalent:

1. R is left-max stable.

2. Every simple left R-module RK has the form K = Uk ∪·{0} for any 0 6= k ∈ K.

Proof. Let RK be a simple left R-module, and let 0 6= k ∈ K. ∗ max (1) ⇒ (2). Given (1), we have K = Uk using Lemma 5.3.1 with L = l(k) ⊆ RR because K is simple. Hence, it follows that K = K∗ ∪·{0} = Uk ∪·{0}, proving (2). (2) ⇒ (1). Given (2), we have Uk ∪·{0} = K = K∗ ∪·{0}. Hence, Uk = K∗. Therefore, (1) follows again by Lemma 5.3.1 because every maximal left ideal L of R arises as L = l(k) for some nonzero element k in a simple left module K, where K =∼ R/L.

Proposition 5.3.10. For any ring R with an ideal I ⊆ J(R), we have:

R is left-max stable if and only if R/I is left-max stable.

Proof. Let X be the left-max idealtor, and let ϕ : R → R/I be the coset map. Then, ϕ is X -full max max and X -ﬁt because L ⊆ RR if and only if L/I ⊆ R/I R/I. Hence, the result follows using Theorem 2.1.6.

Proposition 5.3.11. The left-max idealtor X is not a closed left idealtor.

∼ Proof. Consider the ring of integers Z. Then, clearly 4Z = 2Z. But, we have 2Z ∈ X (Z) and 4Z ∈/ X (Z) by Example 5.3.6. Therefore, X is not a closed left idealtor.

Proposition 5.3.12. The left-max idealtor X is normal.

Proof. Assume that L ∈ X (R), and let u ∈ U(R); we need to show that Lu ∈ X (R). First, we prove the following claim:

Claim. If u ∈ R is a unit, then R/L =∼ R/Lu for every left ideal L of R. Proof. If α : R → R/L is deﬁned by rα = ru−1 + L = r(u−1 + L) for all r ∈ R. Then, α is R-linear, and it is epic because (ru)α = r + L for each r ∈ R. Finally, the kernel of the map α is {r ∈ R | ru−1 ∈ L} = Lu, proving the claim.

Now, if L ∈ X (R) and u is a unit in R, then R/Lu =∼ R/L is simple. Hence, Lu is a maximal left ideal of R and so we have Lu ∈ X (R), proving that X is normal.

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92 Index

L-fit, 25 IC ring,6 L-full, 25 ideal extension, 31 ideal-simple, 68 abelian ring, 13 idempotent,3 Affordability Theorem, 14 internal cancellation for modules,6 affordable ring,4 Artin, E., 22 Kaplansky’s ring,9 Kaplansky, I.,5,9 Bass, H.,2, 22, 35 Kasch, F., 15 Khurana, D.,6 Camillo, V., 23 casilocal ring, 22 Lam, T.Y.,6, 18, 56 Chen, H.,6 left L-gen-stable ring, 16 closed left idealtor, 44 left L-stable ring,4 closure of a left idealtor, 44 left L-Vaserstein ring, 17 cover of a left idealtor, 10 left artinian ring,9 left duo, 55 Dedekind, R.,6 left exchange element, 21 directly finite ring (DF ring),6 left idealtor,3 Dorroh extension, 31 left Kasch ring, 15 Dorroh, J.L., 31 left mininjective ring, 15 left noetherian ring, 62 Ehrlich, G.,6 left P-injective ring, 15 equivalent left idealtors, 11 left perfect ring, 55 essential submodule, 18 left primitive ideal, 61 exchange ring, 10 left primitive ring, 61 Extended Bass Theorem, 22 left quasi-duo rings, 55 extension ring, 31 left socle of a ring, 78 faithful module, 61 left unimodular property, 60 frame of rings, 77 left uniquely generated ring (left UG ring),5 full idempotent, 36 left width of rings, 71 left-max ideal, 59 generalized context-null extension, 42 left-max idealtor, 82 generalized matrix ring, 37 left-max stable ideal, 82 generalized Morita context ring, 37 left-max stable ring, 82 generalized upper triangular matrix ring, 39 lifting,9 Goodearl, K.R., 13 local ring,5

I-ﬁnite ring, 74 module ring, 58

93 Morita context ring, 37 Shift Lemma, 31 Morita invariant property for rings, 36 small submodule,6 Morita, K., 36 split-null extension, 39 stable range n of rings, 35 Neumann, J. von,5,6 stable range 1 of rings,2 nice left idealtor, 52 strongly regular,5 Nicholson, W.K., 21, 64 subdirect product, 63 normal left idealtor, 43 translation invariant property, 17 orthogonal idempotents, 29, 77 Translation Lemma, 17 Pierce decomposition of rings, 77 Pierce, R.S., 77 unit-regular ring,5 potent ring, 21 upper triangular frame of rings, 77 prime ring, 15 Vaserstein, L.N., 17 primitive idempotent, 77 very semisimple module, 64 regular ring,5 right singular ideal, 18 weakly left duo ring, 55 Wedderburn, J.H.M, 22 semi-regular ring, 10 Wedderburn-Artin Theorem, 22 semi-unit-regular ring, 23 Weyl algebra, 62 semilocal ring,9 Weyl, H., 62 semiperfect ring,9 Width Theorem, 70 semiprime ring, 15 Width Theorem (for left quasi-duo rings), 73 semisimple ring,9 Shepherdson, J.C, 36 Yu, H.-P., 55