Projective modules: Wedderburn rings
|
|
- Colin Harmon
- 5 years ago
- Views:
Transcription
1 Projective modules: Wedderburn rings April 10, Wedderburn rings A Wedderburn ring is an artinian ring which has no nonzero nilpotent left ideals. Note that if R has no left ideals I such that I 2 = 0 then I has no nonzero nilpotent left ideals at all. Also, if R has no nonzero nilpotent two-sided ideals then R has no nonzero nilpotent left ideals. The name Wedderburn ring honors the seminal work of JHM Wedderburn, who proved that a finite-dimensional C-algebra which has no nilpotent left ideals must be a finite direct product of of full matrix algebras over C. This was later generalized by E Artin, who showed that a ring which has the descending chain condition on left ideals and which has no nonzero nilpotent left ideals that is, what we now call a Wedderburn ring must be a finite direct product of matrix algebras, each factor defined over some (possibly noncommutative) field. In particular, every simple artinian ring is a matrix ring over a field. Further work by N Jacobson, A Mal cev and others refined this structure theorem to encompass arbitrary artinian rings, using a tool we now call the Jacobson radical. First, we link the structure of Wedderburn rings to our previous work on projective modules. The following theorem says, in essence, that Wedderburn rings are precisely those for which the category of modules is trivial. (That is not to say that the individual modules are trivial!) Theorem 8.1. Suppose that R is ring. The following are equivalent. 1. Every R-module is projective. 2. Every R-module is completely reducible. 3. R is completely reducible. 4. R is a Wedderburn ring. Proof. 1 2: If every module is projective and N < M then the projectivity of M/N implies that the sequence 0 N M M/N 0 splits, whence N is a direct summand of M. 1
2 Conversely, suppose that every module is completely reducible. Given a module M choose a free module F and a surjection 0 K F M 0. The complete reducibility of F implies that F = K M, whence M is projective. 2 3: If every module is completely reducible then R is. Conversely, if R is completely reducible, then it is a sum of simple modules. Hence every free module is also a sum of simple modules, whence also completely reducible. But then every module is a homomorphic image of a free module, whence completely reducible. 3 4: If R is completely reducible then R is a direct sum of minimal left ideals. Since 1 lies in a finite sum, R = R 1 is in fact a finite sum. Since a simple module is artinian a finite sum of simple modules is artinian. Finally, if I is a minimal left ideal then I is a direct summand, hence I = Ie for some idempotent. In particular, I 2 0. Hence R is a Wedderburn ring. Conversely, since R is artinian it contains at least one minimal left ideal I 1, say. Since R contains no nilpotent left ideals I 1 is a direct summand. Say R = I 1 J 1. Again, if J 1 0 then it contains a minimal left ideal I 2, say. Since I 2 is a direct summand of R then it is a direct summand of J 1, by the Modular Law. Say, R = I 1 I 2 J 2. Continuing this way we obtain a descending chain J 1 > J 2 > which can terminate only when some J k = 0. At this point we have decomposed R as a finite direct sum of minimal left ideals. This theorem is the key to proving the Wedderburn-Artin Theorem. To exploit it letr be any ring and consider the ring End R (R). Each element φ End R (R) is determined by its value on 1, since for every x R we have that φ(x) = φ(x 1) = xφ(1). On the other hand, any choice of r R determines an φ End R (R) with φ(1) = r by defining φ(x) = xr. This gives a bijection between R and End R (R). It is straightforward to check that this bijection is an additive. However, composition of endomorphisms corresponds to the reverse of multiplication: if φ r and ψ s then φψ sr, since (φψ)(x) = (xs)r = x(sr). Said another way, End R (R) = R op, where the opposite ring R op is the same as R, except that multiplication is reversed: if is the multiplication in R op then a b = ba, where the latter multiplication is the one defined in R. Left modules for R become right modules for R op, and the other way around. Now consider a Wedderburn ring R. As a left R-module, R is a finite direct sum of simple submodules (minimal left ideals) R j. Number these so that S 1 = R 1 = R2 = = Rk1 = S2 = R k1+1 = R k1+2 = = R k1+k 2 = Thus The S kj j R = S k1 1 Sk2 2 Skn n. are called the isotypic components. 2
3 Lemma 8.2 (Schur s Lemma). If S and T are simple R-modules then any nonzero homomorphism φ Hom R (S, T ) is an isomorphism. Hence if S = T then Hom R (S, T ) = 0, and End R (S) is a (possibly noncommutative) field. Proof. If 0 φ Hom R (S, T ) then ker φ S and im φ 0. Since S and T are simple this implies that ker φ = 0 and im φ = T. That is, φ is an isomorphism. Lemma 8.3. Suppose that M = S k and N = T l are isotypic, with S and T simple. 1. If S = T then HomR (M, N) = End R (M) = M k (F op ), where F = End R (S). Proof. If φ Hom R (M, N) then φ(x 1,..., x k ) = (φ 1 (x 1,..., x k ),..., φ l (x 1,..., x k )), where φ j (x 1,..., x k ) = i φ ij (x i ), where φ ij Hom R (S, T ). Slightly more explicitly, φ ij is the composition of the inclusion of S into the i-th coordinate of M, followed by φ, followed by projection onto the j-th coordinate of N. In particular, if S = T then each φij = 0, whence φ = 0. On the other hand, if S = T and k = l then φ has been represented by a k k matrix with entries in F = End R (S). However, as before when R is not commutative we must multiply matrices on the right: that is to say, we regard the multiplication in F op. Note that if I is a minimal left ideal in a ring R, then the two-sided ideal IR is isotypic, since it is the sum of the left ideals Ix. If Ix 0 then the map y yx is an R-module isomorphism I = Ix. Suppose moreover that I is a direct summand of R, and that J is any left ideal isomorphic to I. Say R = I K and φ is an R-isomorphism from I to J. The composition R R/K φ = I J < R is in End R (R), and so must equal right multiplication by some x R. That is, J = Ix < IR. In particular, if R is a Wedderburn ring then the isotypic components are the 2-sided ideals generated by the minimal left ideals. 3
4 Theorem 8.4 (Wedderburn-Artin Theorem). If R is a Wedderburn ring then R = M k1 (F 1 ) M kn (F n ), where the F j are (possibly noncommutative) fields. More precisely, if S 1,..., S n are the pairwise nonisomorphic minimal left ideals of R, and S j R = S kj j, as R-modules, then as rings we have the isomorphisms R = S 1 R S n R, where S j R = M kj (F j ) and F j = End R (S j ). In particular, a simple artinian ring is a matrix algebra. Conversely, any matrix algebra is a simple artinian ring, and any finite product of matrix algebras is a Wedderburn ring. Proof. We have proven that R op = End R (R) has such a decomposition, with fields F op j. It remains only to observe that M n (F op ) op = M n (F ), under the map which sends each matrix to its transpose. Note a consequence of this theorem: left Wedderburn is the same as right Wedderburn. Let s examine one of the most important examples of a Wedderburn ring, namely the group algebra F G of a finite group G over a commutative field F of characteristic 0. I claim that every F G-module V is completely reducible. This result is called Maschke s Theorem. First of all, since F < F G, V is an F -vector space. Hence if W is a submodule then there is a vector-space splitting f : V W that is, a linear transformation such that f(x) = x for every x W. We use an averaging process to produce an F G-module splitting f: f(x) = 1 g 1 f(gx). This is a F G-module map since if h G and x V f(hx) = 1 g 1 f(ghx) = 1 hh 1 g 1 f(ghx) = h 1 (gh) 1 f(ghx) = h 1 g 1 f(gx) = h f(x). 4
5 Hence if r = h c hh F G then f(rx) = f( h c h hx) = h c h f(hx) = c h h f(x) = r f(x). h Second, it is a splitting, since if x W then so is each gx, whence f(x) = 1 g 1 f(gx) = 1 g 1 gx = x = x. By contrast, if p = char(f ) then F G is never a Wedderburn ring. Indeed, if we set ζ = g then it is straightforward to check that ζ Z(F G) and ζ 2 = 0. Hence I = ζf G is a nonzero nilpotent ideal. 5
Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35
Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 1. Let R be a commutative ring with 1 0. (a) Prove that the nilradical of R is equal to the intersection of the prime
More informationStructure of rings. Chapter Algebras
Chapter 5 Structure of rings 5.1 Algebras It is time to introduce the notion of an algebra over a commutative ring. So let R be a commutative ring. An R-algebra is a ring A (unital as always) together
More informationREPRESENTATION THEORY WEEK 9
REPRESENTATION THEORY WEEK 9 1. Jordan-Hölder theorem and indecomposable modules Let M be a module satisfying ascending and descending chain conditions (ACC and DCC). In other words every increasing sequence
More informationLECTURE NOTES AMRITANSHU PRASAD
LECTURE NOTES AMRITANSHU PRASAD Let K be a field. 1. Basic definitions Definition 1.1. A K-algebra is a K-vector space together with an associative product A A A which is K-linear, with respect to which
More informationExercises on chapter 4
Exercises on chapter 4 Always R-algebra means associative, unital R-algebra. (There are other sorts of R-algebra but we won t meet them in this course.) 1. Let A and B be algebras over a field F. (i) Explain
More informationNOTES ON SPLITTING FIELDS
NOTES ON SPLITTING FIELDS CİHAN BAHRAN I will try to define the notion of a splitting field of an algebra over a field using my words, to understand it better. The sources I use are Peter Webb s and T.Y
More informationREPRESENTATION THEORY, LECTURE 0. BASICS
REPRESENTATION THEORY, LECTURE 0. BASICS IVAN LOSEV Introduction The aim of this lecture is to recall some standard basic things about the representation theory of finite dimensional algebras and finite
More information7 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 characterizations of simple Artinian and semisimple rings. We begin by reminding
More informationALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA
ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND
More informationChapter 1. Wedderburn-Artin Theory
1.1. Basic Terminology and Examples 1 Chapter 1. Wedderburn-Artin Theory Note. Lam states on page 1: Modern ring theory began when J.J.M. Wedderburn proved his celebrated classification theorem for finite
More informationALGEBRA EXERCISES, PhD EXAMINATION LEVEL
ALGEBRA EXERCISES, PhD EXAMINATION LEVEL 1. Suppose that G is a finite group. (a) Prove that if G is nilpotent, and H is any proper subgroup, then H is a proper subgroup of its normalizer. (b) Use (a)
More informationMATH5735 Modules and Representation Theory Lecture Notes
MATH5735 Modules and Representation Theory Lecture Notes Joel Beeren Semester 1, 2012 Contents 1 Why study modules? 4 1.1 Setup............................................. 4 1.2 How do you study modules?.................................
More informationRepresentations of quivers
Representations of quivers Gwyn Bellamy October 13, 215 1 Quivers Let k be a field. Recall that a k-algebra is a k-vector space A with a bilinear map A A A making A into a unital, associative ring. Notice
More information12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold.
12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold. 12.1. Indecomposability of M and the localness of End
More information33 Idempotents and Characters
33 Idempotents and Characters On this day I was supposed to talk about characters but I spent most of the hour talking about idempotents so I changed the title. An idempotent is defined to be an element
More informationRing Theory Problems. A σ
Ring Theory Problems 1. Given the commutative diagram α A σ B β A σ B show that α: ker σ ker σ and that β : coker σ coker σ. Here coker σ = B/σ(A). 2. Let K be a field, let V be an infinite dimensional
More informationA Primer on Homological Algebra
A Primer on Homological Algebra Henry Y Chan July 12, 213 1 Modules For people who have taken the algebra sequence, you can pretty much skip the first section Before telling you what a module is, you probably
More informationStructure Theorem for Semisimple Rings: Wedderburn-Artin
Structure Theorem for Semisimple Rings: Wedderburn-Artin Ebrahim July 4, 2015 This document is a reorganization of some material from [1], with a view towards forging a direct route to the Wedderburn Artin
More informationFILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS.
FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. Let A be a ring, for simplicity assumed commutative. A filtering, or filtration, of an A module M means a descending sequence of submodules M = M 0
More informationTensor Product of modules. MA499 Project II
Tensor Product of modules A Project Report Submitted for the Course MA499 Project II by Subhash Atal (Roll No. 07012321) to the DEPARTMENT OF MATHEMATICS INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI GUWAHATI
More informationMath 250: Higher Algebra Representations of finite groups
Math 250: Higher Algebra Representations of finite groups 1 Basic definitions Representations. A representation of a group G over a field k is a k-vector space V together with an action of G on V by linear
More informationIntroduction to modules
Chapter 3 Introduction to modules 3.1 Modules, submodules and homomorphisms The problem of classifying all rings is much too general to ever hope for an answer. But one of the most important tools available
More informationINFINITE-DIMENSIONAL DIAGONALIZATION AND SEMISIMPLICITY
INFINITE-DIMENSIONAL DIAGONALIZATION AND SEMISIMPLICITY MIODRAG C. IOVANOV, ZACHARY MESYAN, AND MANUEL L. REYES Abstract. We characterize the diagonalizable subalgebras of End(V ), the full ring of linear
More informationAlgebra Homework, Edition 2 9 September 2010
Algebra Homework, Edition 2 9 September 2010 Problem 6. (1) Let I and J be ideals of a commutative ring R with I + J = R. Prove that IJ = I J. (2) Let I, J, and K be ideals of a principal ideal domain.
More informationMATH 326: RINGS AND MODULES STEFAN GILLE
MATH 326: RINGS AND MODULES STEFAN GILLE 1 2 STEFAN GILLE 1. Rings We recall first the definition of a group. 1.1. Definition. Let G be a non empty set. The set G is called a group if there is a map called
More informationLie Algebra Cohomology
Lie Algebra Cohomology Carsten Liese 1 Chain Complexes Definition 1.1. A chain complex (C, d) of R-modules is a family {C n } n Z of R-modules, together with R-modul maps d n : C n C n 1 such that d d
More informationHomework #05, due 2/17/10 = , , , , , Additional problems recommended for study: , , 10.2.
Homework #05, due 2/17/10 = 10.3.1, 10.3.3, 10.3.4, 10.3.5, 10.3.7, 10.3.15 Additional problems recommended for study: 10.2.1, 10.2.2, 10.2.3, 10.2.5, 10.2.6, 10.2.10, 10.2.11, 10.3.2, 10.3.9, 10.3.12,
More informationALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ.
ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ. ANDREW SALCH 1. Hilbert s Nullstellensatz. The last lecture left off with the claim that, if J k[x 1,..., x n ] is an ideal, then
More informationMATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA
MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA These are notes for our first unit on the algebraic side of homological algebra. While this is the last topic (Chap XX) in the book, it makes sense to
More informationCOURSE SUMMARY FOR MATH 504, FALL QUARTER : MODERN ALGEBRA
COURSE SUMMARY FOR MATH 504, FALL QUARTER 2017-8: MODERN ALGEBRA JAROD ALPER Week 1, Sept 27, 29: Introduction to Groups Lecture 1: Introduction to groups. Defined a group and discussed basic properties
More informationTopics in Module Theory
Chapter 7 Topics in Module Theory This chapter will be concerned with collecting a number of results and constructions concerning modules over (primarily) noncommutative rings that will be needed to study
More informationRadical Endomorphisms of Decomposable Modules
Radical Endomorphisms of Decomposable Modules Julius M. Zelmanowitz University of California, Oakland, CA 94607 USA Abstract An element of the Jacobson radical of the endomorphism ring of a decomposable
More informationInjective Modules and Matlis Duality
Appendix A Injective Modules and Matlis Duality Notes on 24 Hours of Local Cohomology William D. Taylor We take R to be a commutative ring, and will discuss the theory of injective R-modules. The following
More informationAlgebras. Chapter Definition
Chapter 4 Algebras 4.1 Definition It is time to introduce the notion of an algebra over a commutative ring. So let R be a commutative ring. An R-algebra is a ring A (unital as always) that is an R-module
More informationCOHEN-MACAULAY RINGS SELECTED EXERCISES. 1. Problem 1.1.9
COHEN-MACAULAY RINGS SELECTED EXERCISES KELLER VANDEBOGERT 1. Problem 1.1.9 Proceed by induction, and suppose x R is a U and N-regular element for the base case. Suppose now that xm = 0 for some m M. We
More informationTori for some locally integral group rings
Tori for some locally integral group rings Diagonalizability over a principal ideal domain Master's Thesis Fabian Hartkopf May 217 Contents Preface............................................... 5.1 Introduction..........................................
More informationInfinite-Dimensional Triangularization
Infinite-Dimensional Triangularization Zachary Mesyan March 11, 2018 Abstract The goal of this paper is to generalize the theory of triangularizing matrices to linear transformations of an arbitrary vector
More informationThe Zero Divisor Conjecture and Self-Injectivity for Monoid Rings
The Zero Divisor Conjecture and Self-Injectivity for Monoid Rings Joe Sullivan May 2, 2011 1 Background 1.1 Monoid Rings Definition 1.1. Let G be a set and let : G G G be a binary operation on G. Then
More informationCourse 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra
Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................
More informationALGEBRA QUALIFYING EXAM, FALL 2017: SOLUTIONS
ALGEBRA QUALIFYING EXAM, FALL 2017: SOLUTIONS Your Name: Conventions: all rings and algebras are assumed to be unital. Part I. True or false? If true provide a brief explanation, if false provide a counterexample
More informationMATH 101A: ALGEBRA I PART C: TENSOR PRODUCT AND MULTILINEAR ALGEBRA. This is the title page for the notes on tensor products and multilinear algebra.
MATH 101A: ALGEBRA I PART C: TENSOR PRODUCT AND MULTILINEAR ALGEBRA This is the title page for the notes on tensor products and multilinear algebra. Contents 1. Bilinear forms and quadratic forms 1 1.1.
More information4.4 Noetherian Rings
4.4 Noetherian Rings Recall that a ring A is Noetherian if it satisfies the following three equivalent conditions: (1) Every nonempty set of ideals of A has a maximal element (the maximal condition); (2)
More informationLINEAR ALGEBRA II: PROJECTIVE MODULES
LINEAR ALGEBRA II: PROJECTIVE MODULES Let R be a ring. By module we will mean R-module and by homomorphism (respectively isomorphism) we will mean homomorphism (respectively isomorphism) of R-modules,
More informationA COURSE IN HOMOLOGICAL ALGEBRA CHAPTER 11: Auslander s Proof of Roiter s Theorem E. L. Lady (April 29, 1998)
A COURSE IN HOMOLOGICAL ALGEBRA CHAPTER 11: Auslander s Proof of Roiter s Theorem E. L. Lady (April 29, 1998) A category C is skeletally small if there exists a set of objects in C such that every object
More informationTopics in Algebra. Dorottya Sziráki Gergő Nemes Mohamed Khaled
Topics in Algebra Dorottya Sziráki Gergő Nemes Mohamed Khaled ii Preface These notes are based on the lectures of the course titled Topics in Algebra, which were given by Mátyás Domokos in the fall trimester
More informationFormal power series rings, inverse limits, and I-adic completions of rings
Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely
More informationSome practice problems for midterm 2
Some practice problems for midterm 2 Kiumars Kaveh November 14, 2011 Problem: Let Z = {a G ax = xa, x G} be the center of a group G. Prove that Z is a normal subgroup of G. Solution: First we prove Z is
More informationGraduate Preliminary Examination
Graduate Preliminary Examination Algebra II 18.2.2005: 3 hours Problem 1. Prove or give a counter-example to the following statement: If M/L and L/K are algebraic extensions of fields, then M/K is algebraic.
More informationNOTES ON LINEAR ALGEBRA OVER INTEGRAL DOMAINS. Contents. 1. Introduction 1 2. Rank and basis 1 3. The set of linear maps 4. 1.
NOTES ON LINEAR ALGEBRA OVER INTEGRAL DOMAINS Contents 1. Introduction 1 2. Rank and basis 1 3. The set of linear maps 4 1. Introduction These notes establish some basic results about linear algebra over
More informationMATH 221 NOTES BRENT HO. Date: January 3, 2009.
MATH 22 NOTES BRENT HO Date: January 3, 2009. 0 Table of Contents. Localizations......................................................................... 2 2. Zariski Topology......................................................................
More informationTENSOR PRODUCTS. (5) A (distributive) multiplication on an abelian group G is a Z-balanced map G G G.
TENSOR PRODUCTS Balanced Maps. Note. One can think of a balanced map β : L M G as a multiplication taking its values in G. If instead of β(l, m) we write simply lm (a notation which is often undesirable)
More information1.5 The Nil and Jacobson Radicals
1.5 The Nil and Jacobson Radicals The idea of a radical of a ring A is an ideal I comprising some nasty piece of A such that A/I is well-behaved or tractable. Two types considered here are the nil and
More informationAlgebra Exam Syllabus
Algebra Exam Syllabus The Algebra comprehensive exam covers four broad areas of algebra: (1) Groups; (2) Rings; (3) Modules; and (4) Linear Algebra. These topics are all covered in the first semester graduate
More informationHARTSHORNE EXERCISES
HARTSHORNE EXERCISES J. WARNER Hartshorne, Exercise I.5.6. Blowing Up Curve Singularities (a) Let Y be the cusp x 3 = y 2 + x 4 + y 4 or the node xy = x 6 + y 6. Show that the curve Ỹ obtained by blowing
More informationNon-commutative Algebra
Non-commutative Algebra Patrick Da Silva Freie Universität Berlin June 2, 2017 Table of Contents 1 Introduction to unital rings 5 1.1 Generalities............................................ 5 1.2 Centralizers
More informationRELATIVE HOMOLOGY. M. Auslander Ø. Solberg
RELATIVE HOMOLOGY M. Auslander Ø. Solberg Department of Mathematics Institutt for matematikk og statistikk Brandeis University Universitetet i Trondheim, AVH Waltham, Mass. 02254 9110 N 7055 Dragvoll USA
More informationLecture 2. (1) Every P L A (M) has a maximal element, (2) Every ascending chain of submodules stabilizes (ACC).
Lecture 2 1. Noetherian and Artinian rings and modules Let A be a commutative ring with identity, A M a module, and φ : M N an A-linear map. Then ker φ = {m M : φ(m) = 0} is a submodule of M and im φ is
More informationSolutions to Assignment 4
1. Let G be a finite, abelian group written additively. Let x = g G g, and let G 2 be the subgroup of G defined by G 2 = {g G 2g = 0}. (a) Show that x = g G 2 g. (b) Show that x = 0 if G 2 = 2. If G 2
More informationThe Cyclic Decomposition Theorem
The Cyclic Decomposition Theorem Math 481/525, Fall 2009 Let V be a finite-dimensional F -vector space, and let T : V V be a linear transformation. In this note we prove that V is a direct sum of cyclic
More informationChapter 5. Linear Algebra
Chapter 5 Linear Algebra The exalted position held by linear algebra is based upon the subject s ubiquitous utility and ease of application. The basic theory is developed here in full generality, i.e.,
More informationALGEBRA QUALIFYING EXAM SPRING 2012
ALGEBRA QUALIFYING EXAM SPRING 2012 Work all of the problems. Justify the statements in your solutions by reference to specific results, as appropriate. Partial credit is awarded for partial solutions.
More informationφ(xy) = (xy) n = x n y n = φ(x)φ(y)
Groups 1. (Algebra Comp S03) Let A, B and C be normal subgroups of a group G with A B. If A C = B C and AC = BC then prove that A = B. Let b B. Since b = b1 BC = AC, there are a A and c C such that b =
More informationRING ELEMENTS AS SUMS OF UNITS
1 RING ELEMENTS AS SUMS OF UNITS CHARLES LANSKI AND ATTILA MARÓTI Abstract. In an Artinian ring R every element of R can be expressed as the sum of two units if and only if R/J(R) does not contain a summand
More informationASSOCIATIEVE ALGEBRA. Eric Jespers. webstek: efjesper HOC: donderdag uur, F.5.207
ASSOCIATIEVE ALGEBRA Eric Jespers 2017 2018 webstek: http://homepages.vub.ac.be/ efjesper HOC: donderdag 09-11 uur, F.5.207 Contents 1 Introduction iii 2 Semisimple rings 1 2.1 Introduction.........................
More informationSOLUTIONS FOR FINITE GROUP REPRESENTATIONS BY PETER WEBB. Chapter 8
SOLUTIONS FOR FINITE GROUP REPRESENTATIONS BY PETER WEBB CİHAN BAHRAN I changed the notation in some of the questions. Chapter 8 1. Prove that if G is any finite group then the only idempotents in the
More informationIntroduction to abstract algebra: definitions, examples, and exercises
Introduction to abstract algebra: definitions, examples, and exercises Travis Schedler January 21, 2015 1 Definitions and some exercises Definition 1. A binary operation on a set X is a map X X X, (x,
More informationIDEAL CLASSES AND RELATIVE INTEGERS
IDEAL CLASSES AND RELATIVE INTEGERS KEITH CONRAD The ring of integers of a number field is free as a Z-module. It is a module not just over Z, but also over any intermediate ring of integers. That is,
More informationINJECTIVE MODULES AND THE INJECTIVE HULL OF A MODULE, November 27, 2009
INJECTIVE ODULES AND THE INJECTIVE HULL OF A ODULE, November 27, 2009 ICHIEL KOSTERS Abstract. In the first section we will define injective modules and we will prove some theorems. In the second section,
More informationAppendix Homework and answers
Appendix Homework and answers Algebra II: Homework These are the weekly homework assignments and answers. Students were encouraged to work on them in groups. Some of the problems turned out to be more
More informationand this makes M into an R-module by (1.2). 2
1. Modules Definition 1.1. Let R be a commutative ring. A module over R is set M together with a binary operation, denoted +, which makes M into an abelian group, with 0 as the identity element, together
More informationALGEBRA HW 4. M 0 is an exact sequence of R-modules, then M is Noetherian if and only if M and M are.
ALGEBRA HW 4 CLAY SHONKWILER (a): Show that if 0 M f M g M 0 is an exact sequence of R-modules, then M is Noetherian if and only if M and M are. Proof. ( ) Suppose M is Noetherian. Then M injects into
More informationArtin algebras of dominant dimension at least 2.
WS 2007/8 Selected Topics CMR Artin algebras of dominant dimension at least 2. Claus Michael Ringel We consider artin algebras with duality functor D. We consider left modules (usually, we call them just
More informationALGEBRA HW 3 CLAY SHONKWILER
ALGEBRA HW 3 CLAY SHONKWILER (a): Show that R[x] is a flat R-module. 1 Proof. Consider the set A = {1, x, x 2,...}. Then certainly A generates R[x] as an R-module. Suppose there is some finite linear combination
More informationAlgebra Qualifying Exam August 2001 Do all 5 problems. 1. Let G be afinite group of order 504 = 23 32 7. a. Show that G cannot be isomorphic to a subgroup of the alternating group Alt 7. (5 points) b.
More informationThe Structure of Hopf Algebras Acting on Galois Extensions
The Structure of Hopf Algebras Acting on Galois Extensions Robert G. Underwood Department of Mathematics and Computer Science Auburn University at Montgomery Montgomery, Alabama June 7, 2016 Abstract Let
More information3.2 Modules of Fractions
3.2 Modules of Fractions Let A be a ring, S a multiplicatively closed subset of A, and M an A-module. Define a relation on M S = { (m, s) m M, s S } by, for m,m M, s,s S, 556 (m,s) (m,s ) iff ( t S) t(sm
More informationNoetherian property of infinite EI categories
Noetherian property of infinite EI categories Wee Liang Gan and Liping Li Abstract. It is known that finitely generated FI-modules over a field of characteristic 0 are Noetherian. We generalize this result
More informationMath 210B:Algebra, Homework 2
Math 210B:Algebra, Homework 2 Ian Coley January 21, 2014 Problem 1. Is f = 2X 5 6X + 6 irreducible in Z[X], (S 1 Z)[X], for S = {2 n, n 0}, Q[X], R[X], C[X]? To begin, note that 2 divides all coefficients
More informationLECTURE 3: REPRESENTATION THEORY OF SL 2 (C) AND sl 2 (C)
LECTURE 3: REPRESENTATION THEORY OF SL 2 (C) AND sl 2 (C) IVAN LOSEV Introduction We proceed to studying the representation theory of algebraic groups and Lie algebras. Algebraic groups are the groups
More informationAssigned homework problems S. L. Kleiman, fall 2008
18.705 Assigned homework problems S. L. Kleiman, fall 2008 Problem Set 1. Due 9/11 Problem R 1.5 Let ϕ: A B be a ring homomorphism. Prove that ϕ 1 takes prime ideals P of B to prime ideals of A. Prove
More information2.3 The Krull-Schmidt Theorem
2.3. THE KRULL-SCHMIDT THEOREM 41 2.3 The Krull-Schmidt Theorem Every finitely generated abelian group is a direct sum of finitely many indecomposable abelian groups Z and Z p n. We will study a large
More informationRadical-theoretic approach to ring theory
Radical-theoretic approach to ring theory 14 th International Workshop for Young Mathematicians Algebra Tomasz Kania Lancaster University: Department of Mathematics and Statistics 10 th -16 th July 2011
More informationEXTENSIONS OF SIMPLE MODULES AND THE CONVERSE OF SCHUR S LEMMA arxiv: v2 [math.ra] 13 May 2009
EXTENSIONS OF SIMPLE MODULES AND THE CONVERSE OF SCHUR S LEMMA arxiv:0903.2490v2 [math.ra] 13 May 2009 GREG MARKS AND MARKUS SCHMIDMEIER Abstract. The converse of Schur s lemma (or CSL) condition on a
More informationarxiv: v1 [math.ag] 2 Oct 2009
THE GENERALIZED BURNSIDE THEOREM IN NONCOMMUTATIVE DEFORMATION THEORY arxiv:09100340v1 [mathag] 2 Oct 2009 EIVIND ERIKSEN Abstract Let A be an associative algebra over a field k, and let M be a finite
More informationCHARACTERS AS CENTRAL IDEMPOTENTS
CHARACTERS AS CENTRAL IDEMPOTENTS CİHAN BAHRAN I have recently noticed (while thinking about the skewed orthogonality business Theo has mentioned) that the irreducible characters of a finite group G are
More informationINSTITUTE OF MATHEMATICS of the Polish Academy of Sciences
INSTITUTE OF MATHEMATICS of the Polish Academy of Sciences ul. Śniadeckich 8, P.O.B. 21, 00-956 Warszawa 10, Poland http://www.impan.pl IM PAN Preprint 722 (2010) Tomasz Maszczyk On Splitting Polynomials
More informationProjective and Injective Modules
Projective and Injective Modules Push-outs and Pull-backs. Proposition. Let P be an R-module. The following conditions are equivalent: (1) P is projective. (2) Hom R (P, ) is an exact functor. (3) Every
More informationA Krull-Schmidt Theorem for Noetherian Modules *
A Krull-Schmidt Theorem for Noetherian Modules * Gary Brookfield Department of Mathematics, University of California, Riverside CA 92521-0135 E-mail: brookfield@math.ucr.edu We prove a version of the Krull-Schmidt
More informationAlgebraic Topology exam
Instituto Superior Técnico Departamento de Matemática Algebraic Topology exam June 12th 2017 1. Let X be a square with the edges cyclically identified: X = [0, 1] 2 / with (a) Compute π 1 (X). (x, 0) (1,
More informationJoseph Muscat Universal Algebras. 1 March 2013
Joseph Muscat 2015 1 Universal Algebras 1 Operations joseph.muscat@um.edu.mt 1 March 2013 A universal algebra is a set X with some operations : X n X and relations 1 X m. For example, there may be specific
More informationMATH 101A: ALGEBRA I PART C: TENSOR PRODUCT AND MULTILINEAR ALGEBRA. This is the title page for the notes on tensor products and multilinear algebra.
MATH 101A: ALGEBRA I PART C: TENSOR PRODUCT AND MULTILINEAR ALGEBRA This is the title page for the notes on tensor products and multilinear algebra. Contents 1. Bilinear forms and quadratic forms 1 1.1.
More informationVector Bundles and Projective Modules. Mariano Echeverria
Serre-Swan Correspondence Serre-Swan Correspondence If X is a compact Hausdorff space the category of complex vector bundles over X is equivalent to the category of finitely generated projective C(X )-modules.
More informationON STRONGLY PRIME IDEALS AND STRONGLY ZERO-DIMENSIONAL RINGS. Christian Gottlieb
ON STRONGLY PRIME IDEALS AND STRONGLY ZERO-DIMENSIONAL RINGS Christian Gottlieb Department of Mathematics, University of Stockholm SE-106 91 Stockholm, Sweden gottlieb@math.su.se Abstract A prime ideal
More informationMath 121 Homework 5: Notes on Selected Problems
Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements
More informationTHE LENGTH OF NOETHERIAN MODULES
THE LENGTH OF NOETHERIAN MODULES GARY BROOKFIELD Abstract. We define an ordinal valued length for Noetherian modules which extends the usual definition of composition series length for finite length modules.
More informationMorita Equivalence. Eamon Quinlan
Morita Equivalence Eamon Quinlan Given a (not necessarily commutative) ring, you can form its category of right modules. Take this category and replace the names of all the modules with dots. The resulting
More informationAzumaya Algebras. Dennis Presotto. November 4, Introduction: Central Simple Algebras
Azumaya Algebras Dennis Presotto November 4, 2015 1 Introduction: Central Simple Algebras Azumaya algebras are introduced as generalized or global versions of central simple algebras. So the first part
More informationCHAPTER I. Rings. Definition A ring R is a set with two binary operations, addition + and
CHAPTER I Rings 1.1 Definitions and Examples Definition 1.1.1. A ring R is a set with two binary operations, addition + and multiplication satisfying the following conditions for all a, b, c in R : (i)
More information8 The Socle and Radical.
8 The Socle and Radical. Simple and semisimple modules are clearly the main building blocks in much of ring theory. Of coure, not every module can be built from semisimple modules, but for many modules
More information5 An Informative Example.
5 An Informative Example. Our primary concern with equivalence is what it involves for module categories. Once we have the Morita characterization of equivalences of module categories, we ll be able to
More information