DOCSLIB.ORG

Π-Complemented Algebras Through Pseudocomplemented Lattices

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Π-Complemented Algebras Through Pseudocomplemented Lattices Order (2012) 29:463–479 DOI 10.1007/s11083-011-9214-4 π-Complemented Algebras Through Pseudocomplemented Lattices Juan Carlos Cabello · Miguel Cabrera · Antonio Fernández López Received: 8 April 2010 / Accepted: 5 April 2011 / Published online: 5 May 2011 © Springer Science+Business Media B.V. 2011 Abstract For an ideal I of a nonassociative algebra A,theπ-closure of I is defined by I = Ann(Ann(I)), where Ann(I) denotes the annihilator of I, i.e., the largest ideal J of A such that IJ = JI = 0. An algebra A is said to be π-complemented if for every π-closed ideal U of A there exists a π-closed ideal V of A such that A = U ⊕ V. For instance, the centrally closed semiprime ring, and the AW∗-algebras (or more generally, boundedly centrally closed C∗-algebras) are π-complemented algebras. In this paper we develop a structure theory for π-complemented algebras by using and revisiting some results of the structure theory for pseudocomplemented lattices. Keywords Pseudocomplemented lattices · Semiprime algebras · Closure operations · Complemented algebras · Decomposable algebras Mathematics Subject Classifications (2010) Primary 17A60; Secondary 06D15 J.C. Cabello and M. Cabrera were supported by the MICINN and Fondos FEDER, MTM2009-12067, and, in addition, by the Junta de Andalucía Grant FQM290. A. Fernández López was supported by the MEC and Fondos FEDER, MTM2007-61978 and MTM2010-19482. J. C. Cabello · M. Cabrera Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain J. C. Cabello e-mail: [email protected] M. Cabrera e-mail: [email protected] A. Fernández López (B) Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain e-mail: [email protected] 464 Order (2012) 29:463–479 1 Introduction Unless otherwise stated, A will denote a not necessarily associative algebra over an arbitrary commutative ring of scalars. Recall that A is said to be complemented if for every ideal I of A there exists an ideal J of A such that A = I ⊕ J. Examples of complemented algebras are the null algebras and the decomposable algebras, where by a null algebra we mean an algebra with zero product, a decomposable algebra is an algebra that is isomorphic to a direct sum of simple algebras, and a simple algebra is a non-null algebra lacking nonzero proper ideals. By regarding any algebra as a left module over its multiplication algebra, the standard characterization of completely reducible modules can be rewritten as follows: A non-null algebra is complemented if, and only if it is the direct sum of a null algebra and a decomposable algebra. Given an algebra A and a closure operation ∼ on the complete lattice IA of all (two-sided) ideals of A, borrowing terminology from the theory of Banach spaces, we will say that a ∼-closed ideal U of A is ∼-complemented (resp. ∼-quasicomplemen- ted)inA if there exists a ∼-closed ideal V of A,calleda∼-complement (resp. ∼-quasicomplement)ofU, such that A = U ⊕ V (resp. A = (U ⊕ V) ∼ ). We say that A is a ∼-complemented (resp. ∼-quasicomplemented) algebra when every ∼-closed ideal of A is ∼-complemented (resp. ∼-quasicomplemented) in A. Note that if ∼ is the discrete closure, i.e., U = U for all U ∈ IA, then complementar- ity, ∼-complementarity and ∼-quasicomplementarity agree. A natural task is the description of the lattice of the ∼-closed ideals. Thus, the study of the lattice of all .-closed ideals and the complementarity in such a lattice has been widely discussed in specific Banach algebras: algebras of functions, and particularly algebras of sequences, (see, e.g., [7, Chapter 4] and [37]); and operator algebras (see, e.g., [30]and[33]). It is worth pointing out that .-quasicomplemented algebras are generalized annihilator normed algebras, which were introduced and studied by B. Yood in the associative context [38], by the third author and A. Rodríguez in the nonassociative setting [15], and revisited in [4]. In the latter paper, the ε-closure was introduced, and proved that the ε-quasicomplemented algebras with zero annihilator are precisely the multiplicativatively semiprime algebras. More recently, in [5], a structure theory for ε-complemented algebras has been achieved. In the present paper we study those algebras which are complemented with respect to the π-closure, i.e., the closure defined on IA by the annihilator operator Ann(.). For each ideal I of A,Ann(I) denotes the largest ideal J of A such that = = π = ( ( )) Iπ IJ JI 0,andthe -closure of I is defined by I Ann Ann I .Theset A of all π-closed ideals of A is a complete lattice for the meet and join operations given by Ui = Ui and Ui = Ui. In section one the concept of annihilator operator on a semilattice is introduced, and a generalization to semilattices with an annihilator operator of a celebrated theorem of Frink and Glivenko [19] for pseudocomplemented semilattices is proved. As a consequence, it is shown that the π-quasicomplemented algebras are precisely the semiprime algebras. In section two we provide a π-structure theory for semiprime algebras: (i) every semiprime algebra is a subdirect product of two semiprime Order (2012) 29:463–479 465 algebras, one of which is π-radical, i.e., it does not contain minimal π-closed ideals, and the other is π-decomposable, i.e., it is the π-closure of the sum of its minimal π-closed ideals, and (ii) π-decomposable semiprime algebras are characterized as essential subdirect products of prime algebras, equivalently, as those semiprime algebras such that the Boolean algebra of their π-closed ideals is atomic; in partic- ular, semiprime algebras satisfying the chain condition on annihilator ideals are π- decomposable. A list of examples of π-decomposable algebras, which runs through different classes of nonassociative algebras, is given at the end of this section. In sec- tion three we discuss pseudocomplemented lattices with additive closure. In section four a structure theory for π-complemented algebras is developed: (i) an algebra is π-complemented if and only it is semiprime and its π-closure is additive, (ii) every π- complemented algebra is the direct sum of a π-radical π-complemented algebra and a π-decomposable π-complemented algebra, (iii) an algebra with zero annihilator coincides with its π-socle if, and only if, it is a direct sum of prime algebras, and (iv) the π-decomposable π-complemented algebras are the hereditary subdirect products of prime algebras. Examples of π-decomposable π-complemented algebras in both functional analysis and pure algebra are collected in this section. Finally, we show that the algebra of all Lebesgue measurable functions on the unit interval is π-radical and π-complemented. 2 Pseudocomplemented Semilattices and Semiprime Algebras We refer to [19], [22]or[16], for basic results on pseudocomplemented semilattices and Boolean algebras. Let (S;∧, 0) be a (meet) semilattice with a least element. By an annihilator operator on S we mean an unary operation x → x∗ of S into itself satisfying the following conditions for all x, y ∈ S (1) x ≤ y ⇒ y∗ ≤ x∗, (2) x ≤ x∗∗, (3) x ∧ y = 0 ⇒ y ≤ x∗. Conditions (1) and (2) say that any annihilator operator defines a symmetric Galois connection.Hence ∗ ∗ ∗ (4) If {xα}⊆S and α xα exists, then α xα exists and α xα = α xα , (5) x∗ = x∗∗∗, and the mapping x → x := x∗∗ is a closure operation on S. By property (5), x ∈ S is closed if, and only if, x = y∗ for some y ∈ S. Denote by B(S) the set of all closed elements of S (also called the skeleton of S [22]). Since x ∧ 0 = 0 for all x ∈ S, it follows from (3) that x ≤ 0∗,so (6) 1 := 0∗ is the largest element both S and B(S). As usual we consider the binary operation ∇:S × S → B(S) defined by x∇ y := (x∗ ∧ y∗)∗. Proposition 2.1 Let (S;∧, 0,∗ ) be a semilattice with annihilator operator. Then (B(S);∇) is a join semilattice with 1 = 0∗ as a largest element and 1∗ as a least element. 466 Order (2012) 29:463–479 Moreover, if S is a lattice, then x∇ y = x ∨ y for all elements x, yand(B(S);∧, ∇) is a lattice. Proof Check that (x, y) → x∇ y defines a join map in B(S). Moreover, if S is a lattice, we have by (4) that x∇ y = x ∨ y and u ∧ v = (u∗ ∨ v∗)∗ ∈ B(S) for all x, y ∈ S and u,v ∈ B(S). Recall that a pseudocomplemented semilattice is a semilattice with a least ele- ment (S;∧, 0) such that for each x ∈ S there exists a largest element x∗ such that x ∧ x∗ = 0. It follows from the basic properties of the pseucomplementation [19]that the pseudocomplemented semilattices are precisely the semilattices with annihilator operator (S;∧, 0,∗ ) such that x ∧ x∗ = 0 for all x ∈ S. A celebrated theorem of Frink and Glivenko proves that if (S;∧, 0,∗ ) is a pseudocomplemented semilattice, then (B(S);∧, ∇,∗ , 0, 1) is a Boolean algebra, i.e., a distributive complemented lattice, with the Boolean complement of an element being its pseudocomplement. This result was proved for complete distributive lattices by V. Glivenko in his early work [20], and was later refined by O. Frink in [17]. For a brief proof of this result, see [27]. Next we will prove an extension of the theorem of Frink and Glivenko for semilattices S with annihilator operator which yields complementation in B(S), but before we show the relationship between pseudocomplemented semilattices and semilattices with an annihilator operator in the lattice of the ideals of a nonassociative algebra.
Load more

Recommended publications
  • Rings Whose Quasi-Injective Modules Are Injecttve K
    proceedings of the american mathematical society Volume 33, Number 2, June 1972 RINGS WHOSE QUASI-INJECTIVE MODULES ARE INJECTTVE K. A. BYRD Abstract. A ring R is called a V-ring, respectively SSI-ring, respectively QH-ring if simple, respectively semisimple, respectively quasi-injective, right Ä-modules are injective. We show that R is SSI if and only if R is a right noetherian V-ring and that any SSI-ring is a finite ring direct sum of simple SSI-rings. We show that if R is left noetherian and SSI then R is QII provided R is hereditary and that in order for R to be hereditary it suffices that maximal right ideals of R be reflexive. An example of Cozzens is cited to show these rings need not be artinian. The rings R which have the property that quasi-injective right R- modules are injective (herein called QII-rings) have been little mentioned in the literature though related classes of rings have been extensively examined. Consider the classes of R-modules consisting of (1) all R- modules, (2) injective R-modules, (3) projective R-modules, (4) quasi- injective R-modules, and (5) quasi-projective R-modules. Of all the nontrivial inclusions one can assume between two of these classes, all but one has been shown to characterize the ring R as either semisimple artinian, quasi-Frobenius, or uniserial. That one excepted is the class of QII-rings whose internal structure, it seems, has not been completely determined. This paper does not attempt a general structure theorem but investi- gates the connection between QII-rings and rings whose semisimple right modules are injective (SSI-rings).
    [Show full text]
  • On Subdirect Product of Prime Modules
    Commun. Korean Math. Soc. 32 (2017), No. 2, pp. 277–285 https://doi.org/10.4134/CKMS.c160103 pISSN: 1225-1763 / eISSN: 2234-3024 ON SUBDIRECT PRODUCT OF PRIME MODULES Najmeh Dehghani and Mohammad Reza Vedadi Abstract. In the various module generalizations of the concepts of prime (semiprime) for a ring, the question “when are semiprime modules sub- direct product of primes?” is a serious question in this context and it is considered by earlier authors in the literature. We continue study on the above question by showing that: If R is Morita equivalent to a right pre-duo ring (e.g., if R is commutative) then weakly compressible R-modules are precisely subdirect products of prime R-modules if and only if dim(R) = 0 and R/N(R) is a semi-Artinian ring if and only if every classical semiprime module is semiprime. In this case, the class of weakly compressible R-modules is an enveloping for Mod-R. Some related conditions are also investigated. 1. Introduction Throughout this paper rings will have a nonzero identity, modules will be right and unitary. In the literature, there are several module generalizations of a semiprime (prime) ring, see [15, Sections 13 and 14] for an excellent reference on the subject. These generalizations introduce various concepts of semiprime (prime) modules and many important theories on semiprime (prime) rings are generalized to modules by them, see; [3], [7], [8], [10] and [17]. The natural question “when are semiprime modules subdirect product of primes?” is then appeared related to these generalizations. In the following, we first recall some definitions of the literature and explain some where the above question was studied.
    [Show full text]
  • Annihilators in Semiprime Right Goldie Rings
    Proceedings of the Edinburgh Mathematical Society (1992) 35, 133-137 © ANNIHILATORS IN SEMIPRIME RIGHT GOLDIE RINGS by C. R. HAJARNAVIS (Received 20th March 1990) (i) Let R be a semiprime right Goldie ring with dim R = n. Then a maximal chain of right annihilators in R has exactly n terms, (ii) A semiprime locally right Goldie ring with ACC and DCC on right annihilators is a right Goldie ring. 1980 Mathematics subject classification (1985 Revision): 16A34. 1. Introduction The aim of this note is to answer two questions raised by Brown and Wehrfritz in [1]. Let R be a semiprime right Goldie ring of uniform dimension n. Since the right singular ideal of R is zero, the right annihilators are complements and so any ascending (or descending) chain of right annihilators in R has at most n terms. While studying certain crossed products, Brown and Wehrfritz have wondered if a maximal such chain necessarily has n terms. This is easy to see if the Goldie chain conditions are assumed to be two-sided. In this case R has a two-sided quotient ring Q of dimension n. The extension of a chain of right annihilators in R can be refined to a chain of length n of right ideals in Q. Intersecting back to R we get a chain of length n in which every term is a right annihilator of a single element of R; this last property following from the fact that elements of Q are expressible as left fractions. We demonstrate here that although the above strategy fails under one-sided assumptions, further analysis of right complements in R shows the required result to be true (Theorem 3.4).
    [Show full text]
  • A Note on Varieties of Semiprime Rings with Semigroup Identities
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 54, 42-45 (1978) A Note on Varieties of Semiprime Rings with Semigroup Identities IGOR Z. GOLUBCHIK AND ALEXANDER MIKHALEV Department of Mechanics and Mathematics, Moscow State University, Moscow V-234, USSR Communicated by A. W. Goldie Received November 14. 1977 Let o(x, ,..., xk) and u(xr ,..., xk) be any monomials, f(x, ,..., x~) = @I ,**., Xt) - u(q >..., xk), and degf the total degree of f(zl ,..., xk). A poly- nomial identityf(x, ,..., x~) = 0 will be called a semigroup identity. Examples of rings with semigroup identities. (1) R is a commutative ring. Then we take f(xl , ~a) = xi.‘cz - x.~x, . (2) R is a finite ring. Indeed (see, for instance, [5]), let I R j = r < 03. Consider fi : R -+ R, i = I,2 ,..., given by fi(a) = ui for all a E R. We have no more than F-I distinct maps fi . So fit = fi for some I < K. Thus ux- = a2 for all a E R and we take f(x) = .G’ - 9. We remark that varieties of the form var(R), where R is a finite ring, were described in many ways among all other varieties of rings by Lvov [5, 61 and Kruse [7]. Our main result is the following THEOREM. Let R be a semiprime associative ring with a semigroup identity. Denote the variety of R by var(R). Then var(R) = var(A @ B), where A is a finite ring and B is a commutative ring.
    [Show full text]
  • IN Herstein Proved That If R Is a Prime Ring and D Is an Inner Derivation Of
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 6, June 1996 A RESULT ON DERIVATIONS TSIU-KWEN LEE AND JER-SHYONG LIN (Communicated by Ken Goodearl) Abstract. Let R be a semiprime ring with a derivation d and let U be a Lie ideal of R, a R. Suppose that ad(u)n =0forallu U,wherenis a fixed positive integer.∈ Then ad(I)=0forIthe ideal of R generatedby[∈ U, U]and if R is 2-torsion free, then ad(U) = 0. Furthermore, R is a subdirect sum of semiprime homomorphic images R1 and R2 with derivations d1 and d2, induced canonically by d, respectively such that ad1(R1) = 0 and the image of U in R2 is commutative (central if R is 2-torsion free), where a denotes the image of a in R1.Moreover,ifU=R,thenad(R) = 0. This gives Bre˘sar’s theorem without the (n 1)!-torsion free assumption on R. − In [8] I. N. Herstein proved that if R is a prime ring and d is an inner derivation of R such that d(x)n =0forallx Rand n a fixed integer, then d =0.In[6] A. Giambruno and I. N. Herstein extended∈ this result to arbitrary derivations in semiprime rings. In [2] L. Carini and A. Giambruno proved that if R is a prime ring with a derivation d such that d(x)n(x) =0forallx U, a Lie ideal of R, then d(U)=0whenRhas no nonzero nil right ideals, char∈R =2andthesame conclusion holds when n(x)=nis fixed and R is a 2-torsion free6 semiprime ring.
    [Show full text]
  • The Radical-Annihilator Monoid of a Ring,” 2016
    The radical-annihilator monoid of a ring Ryan C. Schwiebert 360fly 1000 Town Center Way Ste. 200 Pittsburgh, PA 15317 Center of Ring Theory and its Applications 321 Morton Hall Ohio University Athens, OH 45701 Abstract Kuratowski’s closure-complement problem gives rise to a monoid generated by the closure and complement operations. Consideration of this monoid yielded an interesting classification of topological spaces, and subsequent decades saw further exploration using other set operations. This article is an exploration of a natural analogue in ring theory: a monoid produced by “radical” and “annihila- tor” maps on the set of ideals of a ring. We succeed in characterizing semiprime rings and commutative dual rings by their radical-annihilator monoids, and we determine the monoids for commutative local zero-dimensional (in the sense of Krull dimension) rings. Keywords: annihilators, radicals, Kuratowski closure-complement problem, semiprime rings, dual rings, ordered monoids 2000 MSC: 16D99, 16N60, 06F05 1. Introduction Kuratowski famously showed in [1] that given a subset of a topological space, it is possible to make at most 14 distinct sets from this subset by using the closure and complement operations. This led to the consideration of the “Kura- arXiv:1803.00516v1 [math.RA] 1 Mar 2018 towski monoid” of a space, where the two operations are looked upon as genera- tors for a monoid of functions on the powerset of the space. The monoid formed this way can only be one of six particular types depending on the topology. There are actually two problems concealed here. One is “how many elements does the monoid have for a given space?” and the other is “what is the maximum number of distinct sets one can make with a given subset?” In general, the monoid for a space may be strictly larger than the number of different sets producible from a single input.
    [Show full text]
  • Chains of Semiprime and Prime Ideals in Leavitt Path Algebras 3
    CHAINS OF SEMIPRIME AND PRIME IDEALS IN LEAVITT PATH ALGEBRAS G. ABRAMS, B. GREENFELD, Z. MESYAN, AND K. M. RANGASWAMY Abstract. Semiprime ideals of an arbitrary Leavitt path algebra L are described in terms of their generators. This description is then used to show that the semiprime ideals form a complete sublattice of the lattice of ideals of L, and they enjoy a certain gap property identified by Kaplansky in prime spectra of commutative rings. It is also shown that the totally ordered sets that can be realized as prime spectra of Leavitt path algebras are precisely those that have greatest lower bounds on subchains and enjoy the aforementioned gap property. Finally, it is shown that a conjecture of Kaplansky regarding identifying von Neumann regular rings via their prime factors holds for Leavitt path algebras. 1. Introduction Kaplansky included the following results in two of his monographs from the 1970s, the first of which he attributed to R. Hamsher. (See [11, page 1] and [10, Theorem 11], respectively.) Theorem (K-1): A commutative ring R is von Neumann regular if and only if R is semiprime and every prime factor ring of R is von Neumann regular. Theorem (K-2): If P ( Q are prime ideals of a commutative unital ring R, then there are prime ideals P ′, Q′ of R such that P ⊆ P ′ ( Q′ ⊆ Q and Q′ covers P ′, that is, there are no prime ideals of R strictly between P ′ and Q′. Kaplansky took an interest in extending both theorems to noncommutative rings. In each case, his questions inspired substantial new results in the literature.
    [Show full text]
  • Contributions to the Taxonomy of Rings
    Contributions to the Taxonomy of Rings A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University In partial fulfillment of the requirements for the degree Doctor of Philosophy Henry Chimal-Dzul April 2021 © 2021 Henry Chimal-Dzul. All Rights Reserved. 2 This dissertation titled Contributions to the Taxonomy of Rings by HENRY CHIMAL-DZUL has been approved for the Department of Mathematics and the College of Arts and Sciences by Sergio R. L´opez-Permouth Professor of Mathematics Florenz Plassmann Dean, College of Arts and Sciences 3 Abstract CHIMAL-DZUL, HENRY, Ph.D., April 2021, Mathematics Contributions to the Taxonomy of Rings (139 pp.) Director of Dissertation: Sergio R. L´opez-Permouth The present work aims to contribute to the taxonomy of various classes of non-commutative rings. These include the classes of reversible, reflexive, semicommutative, 2-primal, NI, abelian, Dedekind finite, Armendariz, and McCoy rings. In addition, two new families of rings are introduced, namely right real McCoy and polynomial semicommutative. The first contribution concerns the hierarchy and interconnections of reflexive, abelian, and semicommutative rings in the setting of finite rings. It is shown that a minimal reflexive abelian non-semicommutative ring has order 256 and that the group algebra F2D8 is an example of such a ring. This answers an open question in a paper on a taxonomy of finite 2-primal rings. The second set of contributions includes characterizations of reflexive, 2-primal, weakly 2-primal and NI rings in the setting of Morita context rings. The results on Morita context rings which are reflexive are shown to generalize known characterizations of prime and semiprime Morita context rings.
    [Show full text]
  • New Types of Permuting N-Derivations with Their Applications on Associative Rings
    S S symmetry Article New Types of Permuting n-Derivations with Their Applications on Associative Rings Mehsin Jabel Atteya 1,2 1 Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, UK; [email protected] 2 Department of Mathematics, College of Education, Al-Mustansiriyah University, Baghdad 14022, Iraq Received: 10 November 2019; Accepted: 20 December 2019; Published: 25 December 2019 Abstract: In this article, we introduce new generators of a permuting n-derivations to improve and increase the action of usual derivation. We produce a permuting n-generalized semiderivation, a permuting n-semigeneralized semiderivation, a permuting n-antisemigeneralized semiderivation and a permuting skew n-antisemigeneralized semiderivation of non-empty rings with their applications. Actually, we study the behaviour of those types and present their results of semiprime ring R. Examples of various results have also been included. That is, many of the branches of science such as business, engineering and quantum physics, which used a derivation, have the opportunity to invest them in solving their problems. Keywords: permuting semiderivation; permuting skew n-antisemigeneralized semiderivation; semiprime ring; weak zero-divisor; anticommutative ring; semicommutative ring 1. Introduction Through the 20th century, noncommutative rings have only been issues of systematic study quite recently. Commutative rings, on the contrary, have seemed, though in a covered way, much before, and as with countless theories, it all comes back up to Fermat’s Last Theorem. In 1847, the mathematician Lamé stated an optimal solution of Fermat’s Last Theorem. In dissimilarity to commutative ring theory, which increases from quantity theory, non-commutative ring theory progresses from a notion of Hamilton.
    [Show full text]
  • Noncommutative Ring Theory Notes
    NONCOMMUTATIVE RING THEORY NOTES Contents 1. Notational Conventions & Definitions 1 1.1. Notation 1 1.2. Definitions 1 2. Introduction 1 3. Rings with d.c.c. 7 3.1. Finite Dimensional Algebras 12 4. Nil and Nilpotent Ideals 14 5. Quotient Rings 17 5.1. Ideals in Q(R) 25 6. Division Algebras 30 6.1. Brauer Groups 33 7. Gelfand-Kirillov dimension 38 1. Notational Conventions & Definitions In these notes, R always denotes a ring with 1, unless otherwise stated. 1.1. Notation. subset, proper subset, ideal, right/left ideal, right mod- ule, bimodule 1.2. Definitions. simple, semisimple, prime, semiprime, primitive, semiprim- itive, artinian, noetherian, nil, nilpotent, regular, dedekind-finite, es- sential, uniform, goldie 2. Introduction Definition 2.1. A right R-module, MR, is an abelian group under addition together with a map MR × R ! MR, written (m; r) 7! mr, such that the following hold, for all m; n 2 MR, and all r; s 2 R: (1) (m + n)r = mr + nr (2) m(r + s) = mr + ms (3) m(rs) = (mr)s (4) m · 1 = m 1 2 NONCOMMUTATIVE RING THEORY NOTES If the ring R is understood, we usually drop the subscript and just write M in place of MR. A subgroup N of MR is a submodule if NR ⊆ N. Moreover, if N is a submodule of M, then we can form the factor module M=N in the obvious way. As a set, we have M=N = f m + N j m 2 M g, and the action of R is given by (m + N)r = mr + N.
    [Show full text]
  • Unions of Chains of Primes
    UNIONS OF CHAINS OF PRIMES BE’ERI GREENFELD, LOUIS H. ROWEN, AND UZI VISHNE Abstract. The union of an ascending chain of prime ideals is not always prime. The union of an ascending chain of semi-prime ideals is not always semi-prime. We show that these two properties are independent. We also show that the number of non-prime unions of subchains in a chain of primes in a PI-algebra does not exceed the PI-class minus one, and this bound is tight. 1. Introduction In a commutative ring, the union of a chain of prime ideals is prime, and the union of a chain of semiprime ideals is semiprime. This paper demonstrates and measures the failure of these chain conditions in general. Definition 1.1. A ring has the (semi)prime chain property (denoted P↑ and SP ↑, respectively) if the union of any countable chain of (semi)prime ideals is always (semi)prime.1 The property SP ↑ was recognized by Fisher and Snider [4] as the missing hy- pothesis for Kaplansky’s conjecture on regular rings, and they gave an example of a ring without SP ↑. Our focus is on P↑. The class of rings satisfying P↑ is quite large. An easy exercise shows that every commutative ring satisfies P↑, and the same argument yields that the union of strongly prime ideals is strongly prime (P ⊳R is strongly prime if R/P is a domain). In fact, we have the following result: Proposition 1.2. Every ring R which is a finite module over a central subring, satisfies P↑.
    [Show full text]
  • 17 Semiprime Rings
    17 Semiprime Rings. Our next goal is to characterize those rings whose classical ring of left quotients is semisimple. This requires that we degress briey to discuss the notion of a prime ideal in a non-commutative setting. This, in turn, leads to a generalization of the prime radical from commutative algebra. Let R be an arbitrary ring. An ideal P of R is prime in case for every pair a, b ∈ R,ifaRb P , then a ∈ P or b ∈ P . Of course, if R is commutative, this reduces to the usual denition of primeness. 17.1. Proposition. For an ideal in the ring R the following are equivalent: (a) P is prime; (b) For every pair I,J of left ideals of R, IJ P =⇒ I P or J P ; (c) For every pair I,J of right ideals of R, IJ P =⇒ I P or J P ; (d) For every pair I,J of ideals of R, IJ P =⇒ I P or J P . Proof. (a) =⇒ (d) Let IJ P and suppose that J 6 P . Then there is some b ∈ J \ P . But then aRb P for all a ∈ I, so by (a), a ∈ P for all a ∈ I. (d) =⇒ (c) If I,J are right ideals with IJ P , then RIRJ P ,soby(d)I RI P or J RJ P . (c) =⇒ (b) is similar. (b) =⇒ (a) If aRb P , then RaRb P ,soby(b)a ∈ R or b ∈ R. A ring R is prime if 0 is a prime ideal. Thus, an ideal P is prime i the ring R/P is a prime ring.
    [Show full text]