Chapter One Basic Definitions and Results
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Topics in Module Theory
Chapter 7 Topics in Module Theory This chapter will be concerned with collecting a number of results and construc- tions concerning modules over (primarily) noncommutative rings that will be needed to study group representation theory in Chapter 8. 7.1 Simple and Semisimple Rings and Modules In this section we investigate the question of decomposing modules into \simpler" modules. (1.1) De¯nition. If R is a ring (not necessarily commutative) and M 6= h0i is a nonzero R-module, then we say that M is a simple or irreducible R- module if h0i and M are the only submodules of M. (1.2) Proposition. If an R-module M is simple, then it is cyclic. Proof. Let x be a nonzero element of M and let N = hxi be the cyclic submodule generated by x. Since M is simple and N 6= h0i, it follows that M = N. ut (1.3) Proposition. If R is a ring, then a cyclic R-module M = hmi is simple if and only if Ann(m) is a maximal left ideal. Proof. By Proposition 3.2.15, M =» R= Ann(m), so the correspondence the- orem (Theorem 3.2.7) shows that M has no submodules other than M and h0i if and only if R has no submodules (i.e., left ideals) containing Ann(m) other than R and Ann(m). But this is precisely the condition for Ann(m) to be a maximal left ideal. ut (1.4) Examples. (1) An abelian group A is a simple Z-module if and only if A is a cyclic group of prime order. -
Nilpotent Elements Control the Structure of a Module
Nilpotent elements control the structure of a module David Ssevviiri Department of Mathematics Makerere University, P.O BOX 7062, Kampala Uganda E-mail: [email protected], [email protected] Abstract A relationship between nilpotency and primeness in a module is investigated. Reduced modules are expressed as sums of prime modules. It is shown that presence of nilpotent module elements inhibits a module from possessing good structural properties. A general form is given of an example used in literature to distinguish: 1) completely prime modules from prime modules, 2) classical prime modules from classical completely prime modules, and 3) a module which satisfies the complete radical formula from one which is neither 2-primal nor satisfies the radical formula. Keywords: Semisimple module; Reduced module; Nil module; K¨othe conjecture; Com- pletely prime module; Prime module; and Reduced ring. MSC 2010 Mathematics Subject Classification: 16D70, 16D60, 16S90 1 Introduction Primeness and nilpotency are closely related and well studied notions for rings. We give instances that highlight this relationship. In a commutative ring, the set of all nilpotent elements coincides with the intersection of all its prime ideals - henceforth called the prime radical. A popular class of rings, called 2-primal rings (first defined in [8] and later studied in [23, 26, 27, 28] among others), is defined by requiring that in a not necessarily commutative ring, the set of all nilpotent elements coincides with the prime radical. In an arbitrary ring, Levitzki showed that the set of all strongly nilpotent elements coincides arXiv:1812.04320v1 [math.RA] 11 Dec 2018 with the intersection of all prime ideals, [29, Theorem 2.6]. -
On the Injective Hulls of Semisimple Modules
transactions of the american mathematical society Volume 155, Number 1, March 1971 ON THE INJECTIVE HULLS OF SEMISIMPLE MODULES BY JEFFREY LEVINEC) Abstract. Let R be a ring. Let T=@ieI E(R¡Mt) and rV=V\isI E(R/Mt), where each M¡ is a maximal right ideal and E(A) is the injective hull of A for any A-module A. We show the following: If R is (von Neumann) regular, E(T) = T iff {R/Mt}le, contains only a finite number of nonisomorphic simple modules, each of which occurs only a finite number of times, or if it occurs an infinite number of times, it is finite dimensional over its endomorphism ring. Let R be a ring such that every cyclic Ä-module contains a simple. Let {R/Mi]ie¡ be a family of pairwise nonisomorphic simples. Then E(@ts, E(RIMi)) = T~[¡eIE(R/M/). In the commutative regular case these conditions are equivalent. Let R be a commutative ring. Then every intersection of maximal ideals can be written as an irredundant intersection of maximal ideals iff every cyclic of the form Rlf^\te, Mi, where {Mt}te! is any collection of maximal ideals, contains a simple. We finally look at the relationship between a regular ring R with central idempotents and the Zariski topology on spec R. Introduction. Eckmann and Schopf [2] have shown the existence and unique- ness up to isomorphism of the injective hull of any module. The problem has been to take a specific module and find explicitly its injective hull. -
Lectures on Non-Commutative Rings
Lectures on Non-Commutative Rings by Frank W. Anderson Mathematics 681 University of Oregon Fall, 2002 This material is free. However, we retain the copyright. You may not charge to redistribute this material, in whole or part, without written permission from the author. Preface. This document is a somewhat extended record of the material covered in the Fall 2002 seminar Math 681 on non-commutative ring theory. This does not include material from the informal discussion of the representation theory of algebras that we had during the last couple of lectures. On the other hand this does include expanded versions of some items that were not covered explicitly in the lectures. The latter mostly deals with material that is prerequisite for the later topics and may very well have been covered in earlier courses. For the most part this is simply a cleaned up version of the notes that were prepared for the class during the term. In this we have attempted to correct all of the many mathematical errors, typos, and sloppy writing that we could nd or that have been pointed out to us. Experience has convinced us, though, that we have almost certainly not come close to catching all of the goofs. So we welcome any feedback from the readers on how this can be cleaned up even more. One aspect of these notes that you should understand is that a lot of the substantive material, particularly some of the technical stu, will be presented as exercises. Thus, to get the most from this you should probably read the statements of the exercises and at least think through what they are trying to address. -
7 Rings with Semisimple Generators
7 Rings with Semisimple Generators. It is now quite easy to use Morita to obtain the classical Wedderburn and Artin-Wedderburn charac- terizations of simple Artinian and semisimple rings. We begin by reminding you of a few elementary facts about semisimple modules. Recall that a module RM is simple in case it is a non-zero module with no non-trivial submodules. 7.1. Lemma. [Schur’s Lemma] If M is a simple module and N is a module, then 1. Every non-zero homomorphism M → N is a monomorphism; 2. Every non-zero homomorphism N → M is an epimorphism; 3. If N is simple, then every non-zero homomorphism M → N is an isomorphism; In particular, End(RM) is a division ring. A module RM is semisimple if it is the sum of its simple submodules. Then an easy, but important, characterization of semisimple modules is given in [1] (see Theorem 9.6): 7.2. Proposition. For left R-module RM the following are equivalent: (a) M is semisimple; (b) M is a direct sum of simple submodules; (c) Every submodule of M is a direct summand. Let’s begin with a particularly nice example. Indeed, let D be a division ring. Then the regular module DD is a simple module, and hence, a simple progenerator. In fact, every nitely generated module over D is just a nite dimensional D-vector space, and hence, free. That is, the progenerators for D are simply the non-zero nite dimensional D-vector spaces, and so the rings equivalent to D are nothing more than the n n matrix rings over D. -
ON the LEVITZKI RADICAL of MODULES Nico J. Groenewald And
International Electronic Journal of Algebra Volume 15 (2014) 77-89 ON THE LEVITZKI RADICAL OF MODULES Nico J. Groenewald and David Ssevviiri Received: 5 May 2013; Revised: 15 August 2013 Communicated by Abdullah Harmancı Dedicated to the memory of Professor Efraim P. Armendariz Abstract. In [1] a Levitzki module which we here call an l-prime module was introduced. In this paper we define and characterize l-prime submodules. Let N be a submodule of an R-module M. If p l: N := fm 2 M : every l- system of M containingm meets Ng; p we show that l: N coincides with the intersection L(N) of all l-prime submod- ules of M containing N. We define the Levitzki radical of an R-module M as p L(M) = l: 0. Let β(M), U(M) and Rad(M) be the prime radical, upper nil radical and Jacobson radical of M respectively. In general β(M) ⊆ L(M) ⊆ U(M) ⊆ Rad(M). If R is commutative, β(M) = L(M) = U(M) and if R is left Artinian, β(M) = L(M) = U(M) = Rad(M). Lastly, we show that the class of all l-prime modules RM with RM 6= 0 forms a special class of modules. Mathematics Subject Classification (2010): 16D60, 16N40, 16N60, 16N80, 16S90 Keywords: l-prime submodule; semi l-prime; s-prime submodule; upper nil radical; Levitzki radical 1. Introduction All modules are left modules, the rings are associative but not necessarily unital. By I/R and N ≤ M we respectively mean I is an ideal of a ring R and N is a submodule of a module M. -
Virtually Semisimple Modules and a Generalization of the Wedderburn
Virtually Semisimple Modules and a Generalization of the Wedderburn-Artin Theorem ∗†‡ M. Behboodia,b,§ A. Daneshvara and M. R. Vedadia aDepartment of Mathematical Sciences, Isfahan University of Technology P.O.Box : 84156-83111, Isfahan, Iran bSchool of Mathematics, Institute for Research in Fundamental Sciences (IPM) P.O.Box : 19395-5746, Tehran, Iran [email protected] [email protected] [email protected] Abstract By any measure, semisimple modules form one of the most important classes of mod- ules and play a distinguished role in the module theory and its applications. One of the most fundamental results in this area is the Wedderburn-Artin theorem. In this paper, we establish natural generalizations of semisimple modules and give a generalization of the Wedderburn-Artin theorem. We study modules in which every submodule is isomorphic to a direct summand and name them virtually semisimple modules. A module RM is called completely virtually semisimple if each submodules of M is a virtually semisimple module. A ring R is then called left (completely) vir- tually semisimple if RR is a left (compleatly) virtually semisimple R-module. Among other things, we give several characterizations of left (completely) virtually semisimple arXiv:1603.05647v2 [math.RA] 31 May 2016 rings. For instance, it is shown that a ring R is left completely virtually semisimple if ∼ k and only if R = Qi=1 Mni (Di) where k,n1, ..., nk ∈ N and each Di is a principal left ideal domain. Moreover, the integers k, n1, ..., nk and the principal left ideal domains D1, ..., Dk are uniquely determined (up to isomorphism) by R. -
Fully Idempotent and Multiplication Modules Nil ORHAN ERTA¸S
Palestine Journal of Mathematics Vol. 3(Spec 1) (2014) , 432–437 © Palestine Polytechnic University-PPU 2014 Fully Idempotent and Multiplication Modules Nil ORHAN ERTA¸S This paper is dedicated to Professors P.F. Smith and J. Clark on their 70th birthdays . Communicated by Ahmet Sinan Cevik MSC 2010 Classifications: 16D40, 16D80, 13A15. Keywords and phrases: Multiplication module, idempotent submodule. Abstract. A submodule N of M is idempotent if N = N ?N = Hom(M; N)N. In this paper we give some properties of idempotent submodules. Relations between the multiplication, pure, and idempotent submodules are investigated. We give necessary condition for tensor products of two idempotent submodules to be idempotent. 1 Introduction All rings are associative with identity element and all modules are unitary right R-modules. Recall that [N : M] = fr 2 R : Mr ⊆ Ng. r(M) is the annihilator ideal of M in R, i.e. the ideal consisting of all elements x of R such that mx = 0 for all m 2 M. M is said to be faithful, if r(M) = 0. When we generalize notions of ring to module, some difficulties come up with the multiplication in a module. In [5], a product on the lattice of submodules of a module was defined. Let M be an R-module and N and L submodules of M. Set: X N?L := Hom(M; L)N = ff(N) j f : M ! Lg N is called idempotent submodule of M if N?N = N. That is, N is idempotent submodule of M, if for each element n 2 N there exist a positive integer k, homomorphisms 'i : M ! N(1 ≤ i ≤ k) and elements ni 2 N(1 ≤ i ≤ k) such that n = '1(n1) + ··· + 'k(nk). -
Associatieve Algebra
ASSOCIATIEVE ALGEBRA Eric Jespers 2019{2020 webstek: http://homepages.vub.ac.be/sefjesper HOC: donderdag 09-11 uur, G.6.322 Contents 1 Introduction iii 2 Semisimple rings 1 2.1 Introduction . 1 2.2 Ascending and descending chain conditions . 2 2.3 Semisimplicity . 6 2.4 Exercises . 13 2.5 Structure of semisimple rings . 15 2.6 Examples . 21 2.7 Exercises . 23 3 The Jacobson radical 25 3.1 The Jacobson radical . 25 3.2 Nilpotent and nil ideals . 28 3.3 Examples . 30 3.4 Hopkins-Levitzki and Nakayama results . 35 3.5 Von Neumann Regular Rings . 37 3.6 Linear groups and the Burnside problem . 39 3.7 Exercises . 44 4 Prime and primitive rings 47 4.1 Prime rings . 47 4.2 Examples . 51 4.3 Some properties and applications . 54 4.4 Primitive rings . 57 4.5 Examples of primitive rings . 59 4.6 Density Theorem . 66 4.7 Exercises . 75 i ii CONTENTS 5 Skew fields 77 5.1 Wedderburn's Theorem . 77 5.2 Additive commutators . 79 5.3 Commutativity Theorems . 82 5.4 Algebraic Skew Fields . 84 5.5 Exercises . 86 6 Goldie Theorems 87 6.1 Ore Localisations . 87 6.2 Properties of Localisations . 94 6.3 Annihilators . 96 6.4 Goldie rings . 98 6.5 Exercises . 102 Index 106 Chapter 1 Introduction In earlier courses we have introduced important algebraic structures, such as semigroups, groups and rings. We have seen that via represen- tation theory of groups and group rings there is a strong link between group and ring theory. -
Finite Group Representations for the Pure Mathematician
Finite Group Representations for the Pure Mathematician by Peter Webb Preface This book started as notes for courses given at the graduate level at the University of Minnesota. It is intended to be used at the level of a second-year graduate course in an American university. At this level the book should provide material for a year-long course in representation theory. It is supposed that the reader has already studied the material in a first-year graduate course on algebra and is familiar with the basic properties, for example, of Sylow subgroups and solvable groups, as well as the examples which are introduced in a first group theory course, such as the dihedral, symmetric, alternating and quaternion groups. The reader should also be familiar with other aspects of algebra which appear in or before a first- year graduate course, such as Galois Theory, tensor products, Noetherian properties of commutative rings, the structure of modules over a principal ideal domain, and the first properties of ideals. The Pure Mathematician for whom this course is intended may well have a primary in- terest in an area of pure mathematics other than the representation theory of finite groups. Group representations arise naturally in many areas, such as number theory, combinatorics and topology, to name just three, and the aim of this course is to give students in a wide range of areas the technique to understand the representations which they encounter. This point of view has determined to a large extent the nature of this book: it should be suffi- ciently short, so that students who are not specialists in group representations can get to the end of it. -
9. Simple and Semisimple Rings 9.1. Semisimple Rings. a Ring Is
9. Simple and semisimple rings (Apres` Bourbaki Algebra Chapter 8 §5) [s:sssrings] 9.1. Semisimple rings. A ring is semisimple if it is semisimple as a (left) module [ss:ssrings] over itself, or, equivalently, if every (left) module over it is semisimple. (Every module is a sum of its cyclic submodules. And every cyclic module is a quotient of AA.) Some facts about semisimple rings: • A semisimple ring has finite length: in other words, it is Artinian and Noetherian. (Any finitely generated semisimple module has finite length.) • Simple modules of a semisimple ring are precisely its minimal left ideals. (Simple modules are quotients of the ring, but the ring being semisimple every quotient is isomorphic to a sub.) • For any left ideal l of a semisimple ring A, there is an idempotent e such that l = le = Ae. (Write A = l⊕l0. The projection to l is given by right multiplication by an element e. We have e2 = 1e2 = 1e = e and l = Ae = le.) • The isotypic components of a semisimple ring are precisely its minimal two- sided ideals. Any two sided ideal is a sum of these. (Characteristic left ideals are precisely two-sided ideals. Thus two-sided ideals are sums of isotypic components. Conversely, isotypic components being characteristic are two-sided ideals.) • Any quotient of a semisimple ring is semisimple. [p:oppssring] Proposition 9.1. The opposite of a semisimple ring is semisimple. Proof. The opposite of a semisimple module being semisimple (Theorem ??), we know that M opp is semisimple where M denotes a semisimple ring as a left module over itself. -
Noncomrnutative Algebra
Graduate Texts in Mathematics 144 Editorial Board 1. H. Ewing F. W. Gehring P. R. Halmos Graduate Texts in Mathematics TAKEUTI/ZARING. Inlroduction to Axiomatic Set Theory. 2nd ed. 2 OXTOIlY. Measure and Category. 2nd ed. 3 SCHAEFfER. Topological Vector Spaces. 4 HILTON/STAMMBACH. A Course in Homological Algebra. 5 MAC LANE. Categories for the Working Mathematician. 6 HUGHES/PIPER. Projective Planes. 7 SERRE. A Course in Arithmetic. 8 TAKEUTt/ZARING. Axiometic Set Theory. 9 HUMPHREYS. Introduction to Lie Algebras and Representation Theory. 10 COHEN. A Course in Simple Homotopy Theory. 11 CONWAY. Functions of One Complex Variable. 2nd ed. 12 BEALS. Advanced Mathematical Analysis. 13 ANDERSON/FuLLER. Rings and Categories of Modules. 2nd ed. 14 GOLUBITSKy/GUILEMIN. Stable Mappings and Their Singularities. 15 BERBERIAN. Lectures in Functional Analysis and Operator Theory. 16 WINTER. The Structure of Fields. 17 ROSENIlLATT. Random Processes. 2nd ed. 18 HALMos. Measure Theory. 19 HALMOS. A Hilbert Space Problem Book. 2nd ed. 20 HUSEMOLLER. Fibre Bundles. 2nd ed. 21 HUMPHREYS. Linear Algebraic Groups. 22 BARNES/MACK. An Algebraic Introduction to Mathematical Logic. 23 GREUB. Linear Algebra. 4th ed. 24 HOLMES. Geometric Functional Analysis and Its Applications. 25 HEWITTISTROMBERG. Real and Abstract Analysis. 26 MANES. Algebraic Theories. 27 KELLEY. General Topology. 28 ZARISKI/SAMUEL. Commutative Algebra. Vol. I. 29 ZARISKI/SAMUEL. Commutative Algebra. Vol. II. 30 JACOIlSON. Lectures in Abstract Algebra I. Basic Concepts. 31 JACOBSON. Lectures in Abstract Algebra II. Linear Algebra. 32 JACOBSON. Lectures in Abstract Algebra III. Theory of Fields and Galois Theory. 33 HIRSCH. Differential Topology. 34 SPITZER. Principles of Random Walk.