INJECTIVE MODULES OVER a PRTNCIPAL LEFT and RIGHT IDEAL DOMAIN, \Ryith APPLICATIONS
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Injective Modules: Preparatory Material for the Snowbird Summer School on Commutative Algebra
INJECTIVE MODULES: PREPARATORY MATERIAL FOR THE SNOWBIRD SUMMER SCHOOL ON COMMUTATIVE ALGEBRA These notes are intended to give the reader an idea what injective modules are, where they show up, and, to a small extent, what one can do with them. Let R be a commutative Noetherian ring with an identity element. An R- module E is injective if HomR( ;E) is an exact functor. The main messages of these notes are − Every R-module M has an injective hull or injective envelope, de- • noted by ER(M), which is an injective module containing M, and has the property that any injective module containing M contains an isomorphic copy of ER(M). A nonzero injective module is indecomposable if it is not the direct • sum of nonzero injective modules. Every injective R-module is a direct sum of indecomposable injective R-modules. Indecomposable injective R-modules are in bijective correspondence • with the prime ideals of R; in fact every indecomposable injective R-module is isomorphic to an injective hull ER(R=p), for some prime ideal p of R. The number of isomorphic copies of ER(R=p) occurring in any direct • sum decomposition of a given injective module into indecomposable injectives is independent of the decomposition. Let (R; m) be a complete local ring and E = ER(R=m) be the injec- • tive hull of the residue field of R. The functor ( )_ = HomR( ;E) has the following properties, known as Matlis duality− : − (1) If M is an R-module which is Noetherian or Artinian, then M __ ∼= M. -
Varieties of Quasigroups Determined by Short Strictly Balanced Identities
Czechoslovak Mathematical Journal Jaroslav Ježek; Tomáš Kepka Varieties of quasigroups determined by short strictly balanced identities Czechoslovak Mathematical Journal, Vol. 29 (1979), No. 1, 84–96 Persistent URL: http://dml.cz/dmlcz/101580 Terms of use: © Institute of Mathematics AS CR, 1979 Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz Czechoslovak Mathematical Journal, 29 (104) 1979, Praha VARIETIES OF QUASIGROUPS DETERMINED BY SHORT STRICTLY BALANCED IDENTITIES JAROSLAV JEZEK and TOMAS KEPKA, Praha (Received March 11, 1977) In this paper we find all varieties of quasigroups determined by a set of strictly balanced identities of length ^ 6 and study their properties. There are eleven such varieties: the variety of all quasigroups, the variety of commutative quasigroups, the variety of groups, the variety of abelian groups and, moreover, seven varieties which have not been studied in much detail until now. In Section 1 we describe these varieties. A survey of some significant properties of arbitrary varieties is given in Section 2; in Sections 3, 4 and 5 we assign these properties to the eleven varieties mentioned above and in Section 6 we give a table summarizing the results. 1. STRICTLY BALANCED QUASIGROUP IDENTITIES OF LENGTH £ 6 Quasigroups are considered as universal algebras with three binary operations •, /, \ (the class of all quasigroups is thus a variety). -
Dedekind Domains
Dedekind Domains Mathematics 601 In this note we prove several facts about Dedekind domains that we will use in the course of proving the Riemann-Roch theorem. The main theorem shows that if K=F is a finite extension and A is a Dedekind domain with quotient field F , then the integral closure of A in K is also a Dedekind domain. As we will see in the proof, we need various results from ring theory and field theory. We first recall some basic definitions and facts. A Dedekind domain is an integral domain B for which every nonzero ideal can be written uniquely as a product of prime ideals. Perhaps the main theorem about Dedekind domains is that a domain B is a Dedekind domain if and only if B is Noetherian, integrally closed, and dim(B) = 1. Without fully defining dimension, to say that a ring has dimension 1 says nothing more than nonzero prime ideals are maximal. Moreover, a Noetherian ring B is a Dedekind domain if and only if BM is a discrete valuation ring for every maximal ideal M of B. In particular, a Dedekind domain that is a local ring is a discrete valuation ring, and vice-versa. We start by mentioning two examples of Dedekind domains. Example 1. The ring of integers Z is a Dedekind domain. In fact, any principal ideal domain is a Dedekind domain since a principal ideal domain is Noetherian integrally closed, and nonzero prime ideals are maximal. Alternatively, it is easy to prove that in a principal ideal domain, every nonzero ideal factors uniquely into prime ideals. -
Unofficial Math 501 Problems (PDF)
Uno±cial Math 501 Problems The following document consists of problems collected from previous classes and comprehensive exams given at the University of Illinois at Urbana-Champaign. In the exercises below, all rings contain identity and all modules are unitary unless otherwise stated. Please report typos, sug- gestions, gripes, complaints, and criticisms to: [email protected] 1) Let R be a commutative ring with identity and let I and J be ideals of R. If the R-modules R=I and R=J are isomorphic, prove I = J. Note, we really do mean equals! 2) Let R and S be rings where we assume that R has an identity element. If M is any (R; S)-bimodule, prove that S HomR(RR; M) ' M: 3) Let 0 ! A ! B ! C ! 0 be an exact sequence of R-modules where R is a ring with identity. If A and C are projective, show that B is projective. 4) Let R be a ring with identity. (a) Prove that every ¯nitely generated right R-module is noetherian if and only if R is a right noetherian ring. (b) Give an example of a ¯nitely generated module which is not noetherian. ® ¯ 5) Let 0 ! A ! B ! C ! 0 be a short exact sequence of R-modules where R is any ring. Assume there exists an R-module homomorphism ± : B ! A such that ±® = idA. Prove that there exists an R-module homomorphism : C ! B such that ¯ = idC . 6) If R is any ring and RM is an R-module, prove that the functor ¡ R M is right exact. -
Noncommutative Unique Factorization Domainso
NONCOMMUTATIVE UNIQUE FACTORIZATION DOMAINSO BY P. M. COHN 1. Introduction. By a (commutative) unique factorization domain (UFD) one usually understands an integral domain R (with a unit-element) satisfying the following three conditions (cf. e.g. Zariski-Samuel [16]): Al. Every element of R which is neither zero nor a unit is a product of primes. A2. Any two prime factorizations of a given element have the same number of factors. A3. The primes occurring in any factorization of a are completely deter- mined by a, except for their order and for multiplication by units. If R* denotes the semigroup of nonzero elements of R and U is the group of units, then the classes of associated elements form a semigroup R* / U, and A1-3 are equivalent to B. The semigroup R*jU is free commutative. One may generalize the notion of UFD to noncommutative rings by taking either A-l3 or B as starting point. It is obvious how to do this in case B, although the class of rings obtained is rather narrow and does not even include all the commutative UFD's. This is indicated briefly in §7, where examples are also given of noncommutative rings satisfying the definition. However, our principal aim is to give a definition of a noncommutative UFD which includes the commutative case. Here it is better to start from A1-3; in order to find the precise form which such a definition should take we consider the simplest case, that of noncommutative principal ideal domains. For these rings one obtains a unique factorization theorem simply by reinterpreting the Jordan- Holder theorem for right .R-modules on one generator (cf. -
REPRESENTATION THEORY WEEK 9 1. Jordan-Hölder Theorem And
REPRESENTATION THEORY WEEK 9 1. Jordan-Holder¨ theorem and indecomposable modules Let M be a module satisfying ascending and descending chain conditions (ACC and DCC). In other words every increasing sequence submodules M1 ⊂ M2 ⊂ ... and any decreasing sequence M1 ⊃ M2 ⊃ ... are finite. Then it is easy to see that there exists a finite sequence M = M0 ⊃ M1 ⊃···⊃ Mk = 0 such that Mi/Mi+1 is a simple module. Such a sequence is called a Jordan-H¨older series. We say that two Jordan H¨older series M = M0 ⊃ M1 ⊃···⊃ Mk = 0, M = N0 ⊃ N1 ⊃···⊃ Nl = 0 ∼ are equivalent if k = l and for some permutation s Mi/Mi+1 = Ns(i)/Ns(i)+1. Theorem 1.1. Any two Jordan-H¨older series are equivalent. Proof. We will prove that if the statement is true for any submodule of M then it is true for M. (If M is simple, the statement is trivial.) If M1 = N1, then the ∼ statement is obvious. Otherwise, M1 + N1 = M, hence M/M1 = N1/ (M1 ∩ N1) and ∼ M/N1 = M1/ (M1 ∩ N1). Consider the series M = M0 ⊃ M1 ⊃ M1∩N1 ⊃ K1 ⊃···⊃ Ks = 0, M = N0 ⊃ N1 ⊃ N1∩M1 ⊃ K1 ⊃···⊃ Ks = 0. They are obviously equivalent, and by induction assumption the first series is equiv- alent to M = M0 ⊃ M1 ⊃ ··· ⊃ Mk = 0, and the second one is equivalent to M = N0 ⊃ N1 ⊃···⊃ Nl = {0}. Hence they are equivalent. Thus, we can define a length l (M) of a module M satisfying ACC and DCC, and if M is a proper submodule of N, then l (M) <l (N). -
SOME ALGEBRAIC DEFINITIONS and CONSTRUCTIONS Definition
SOME ALGEBRAIC DEFINITIONS AND CONSTRUCTIONS Definition 1. A monoid is a set M with an element e and an associative multipli- cation M M M for which e is a two-sided identity element: em = m = me for all m M×. A−→group is a monoid in which each element m has an inverse element m−1, so∈ that mm−1 = e = m−1m. A homomorphism f : M N of monoids is a function f such that f(mn) = −→ f(m)f(n) and f(eM )= eN . A “homomorphism” of any kind of algebraic structure is a function that preserves all of the structure that goes into the definition. When M is commutative, mn = nm for all m,n M, we often write the product as +, the identity element as 0, and the inverse of∈m as m. As a convention, it is convenient to say that a commutative monoid is “Abelian”− when we choose to think of its product as “addition”, but to use the word “commutative” when we choose to think of its product as “multiplication”; in the latter case, we write the identity element as 1. Definition 2. The Grothendieck construction on an Abelian monoid is an Abelian group G(M) together with a homomorphism of Abelian monoids i : M G(M) such that, for any Abelian group A and homomorphism of Abelian monoids−→ f : M A, there exists a unique homomorphism of Abelian groups f˜ : G(M) A −→ −→ such that f˜ i = f. ◦ We construct G(M) explicitly by taking equivalence classes of ordered pairs (m,n) of elements of M, thought of as “m n”, under the equivalence relation generated by (m,n) (m′,n′) if m + n′ = −n + m′. -
Right Ideals of a Ring and Sublanguages of Science
RIGHT IDEALS OF A RING AND SUBLANGUAGES OF SCIENCE Javier Arias Navarro Ph.D. In General Linguistics and Spanish Language http://www.javierarias.info/ Abstract Among Zellig Harris’s numerous contributions to linguistics his theory of the sublanguages of science probably ranks among the most underrated. However, not only has this theory led to some exhaustive and meaningful applications in the study of the grammar of immunology language and its changes over time, but it also illustrates the nature of mathematical relations between chunks or subsets of a grammar and the language as a whole. This becomes most clear when dealing with the connection between metalanguage and language, as well as when reflecting on operators. This paper tries to justify the claim that the sublanguages of science stand in a particular algebraic relation to the rest of the language they are embedded in, namely, that of right ideals in a ring. Keywords: Zellig Sabbetai Harris, Information Structure of Language, Sublanguages of Science, Ideal Numbers, Ernst Kummer, Ideals, Richard Dedekind, Ring Theory, Right Ideals, Emmy Noether, Order Theory, Marshall Harvey Stone. §1. Preliminary Word In recent work (Arias 2015)1 a line of research has been outlined in which the basic tenets underpinning the algebraic treatment of language are explored. The claim was there made that the concept of ideal in a ring could account for the structure of so- called sublanguages of science in a very precise way. The present text is based on that work, by exploring in some detail the consequences of such statement. §2. Introduction Zellig Harris (1909-1992) contributions to the field of linguistics were manifold and in many respects of utmost significance. -
Noncommutative Localization in Algebra and Topology
Noncommutative localization in algebra and topology ICMS Edinburgh 2002 Edited by Andrew Ranicki Electronic version of London Mathematical Society Lecture Note Series 330 Cambridge University Press (2006) Contents Dedication . vii Preface . ix Historical Perspective . x Conference Participants . xi Conference Photo . .xii Conference Timetable . xiii On atness and the Ore condition J. A. Beachy ......................................................1 Localization in general rings, a historical survey P. M. Cohn .......................................................5 Noncommutative localization in homotopy theory W. G. Dwyer . 24 Noncommutative localization in group rings P. A. Linnell . 40 A non-commutative generalisation of Thomason's localisation theorem A. Neeman . 60 Noncommutative localization in topology A. A. Ranicki . 81 v L2-Betti numbers, Isomorphism Conjectures and Noncommutative Lo- calization H. Reich . 103 Invariants of boundary link cobordism II. The Blanch¯eld-Duval form D. Sheiham . 143 Noncommutative localization in noncommutative geometry Z. Skoda· ........................................................220 vi Dedicated to the memory of Desmond Sheiham (13th November 1974 ¡ 25th March 2005) ² Cambridge University (Trinity College), 1993{1997 B.A. Hons. Mathematics 1st Class, 1996 Part III Mathematics, Passed with Distinction, 1997 ² University of Edinburgh, 1997{2001 Ph.D. Invariants of Boundary Link Cobordism, 2001 ² Visiting Assistant Professor, Mathematics Department, University of California at Riverside, 2001{2003 ² Research Instructor, International University Bremen (IUB), 2003{2005 vii Publications: 1. Non-commutative Characteristic Polynomials and Cohn Localization Journal of the London Mathematical Society (2) Vol. 64, 13{28 (2001) http://arXiv.org/abs/math.RA/0104158 2. Invariants of Boundary Link Cobordism Memoirs of the American Mathematical Society, Vol. 165 (2003) http://arXiv.org/abs/math.AT/0110249 3. Whitehead Groups of Localizations and the Endomorphism Class Group Journal of Algebra, Vol. -
Modular Representation Theory of Finite Groups Jun.-Prof. Dr
Modular Representation Theory of Finite Groups Jun.-Prof. Dr. Caroline Lassueur TU Kaiserslautern Skript zur Vorlesung, WS 2020/21 (Vorlesung: 4SWS // Übungen: 2SWS) Version: 23. August 2021 Contents Foreword iii Conventions iv Chapter 1. Foundations of Representation Theory6 1 (Ir)Reducibility and (in)decomposability.............................6 2 Schur’s Lemma...........................................7 3 Composition series and the Jordan-Hölder Theorem......................8 4 The Jacobson radical and Nakayama’s Lemma......................... 10 Chapter5 2.Indecomposability The Structure of and Semisimple the Krull-Schmidt Algebras Theorem ...................... 1115 6 Semisimplicity of rings and modules............................... 15 7 The Artin-Wedderburn structure theorem............................ 18 Chapter8 3.Semisimple Representation algebras Theory and their of Finite simple Groups modules ........................ 2226 9 Linear representations of finite groups............................. 26 10 The group algebra and its modules............................... 29 11 Semisimplicity and Maschke’s Theorem............................. 33 Chapter12 4.Simple Operations modules on over Groups splitting and fields Modules............................... 3436 13 Tensors, Hom’s and duality.................................... 36 14 Fixed and cofixed points...................................... 39 Chapter15 5.Inflation, The Mackey restriction Formula and induction and Clifford................................ Theory 3945 16 Double cosets........................................... -
Injective Modules and Divisible Groups
University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Masters Theses Graduate School 5-2015 Injective Modules And Divisible Groups Ryan Neil Campbell University of Tennessee - Knoxville, [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes Part of the Algebra Commons Recommended Citation Campbell, Ryan Neil, "Injective Modules And Divisible Groups. " Master's Thesis, University of Tennessee, 2015. https://trace.tennessee.edu/utk_gradthes/3350 This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a thesis written by Ryan Neil Campbell entitled "Injective Modules And Divisible Groups." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Master of Science, with a major in Mathematics. David Anderson, Major Professor We have read this thesis and recommend its acceptance: Shashikant Mulay, Luis Finotti Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) Injective Modules And Divisible Groups AThesisPresentedforthe Master of Science Degree The University of Tennessee, Knoxville Ryan Neil Campbell May 2015 c by Ryan Neil Campbell, 2015 All Rights Reserved. ii Acknowledgements I would like to thank Dr. -
Block Theory with Modules
JOURNAL OF ALGEBRA 65, 225--233 (1980) Block Theory with Modules J. L. ALPERIN* AND DAVID W. BURRY* Department of Mathematics, University of Chicago, Chicago, Illinois 60637 and Department of Mathematics, Yale University, New Haven, Connecticut 06520 Communicated by Walter Felt Received October 2, 1979 1. INTRODUCTION There are three very different approaches to block theory: the "functional" method which deals with characters, Brauer characters, Carton invariants and the like; the ring-theoretic approach with heavy emphasis on idempotents; module-theoretic methods based on relatively projective modules and the Green correspondence. Each approach contributes greatly and no one of them is sufficient for all the results. Recently, a number of fundamental concepts and results have been formulated in module-theoretic terms. These include Nagao's theorem [3] which implies the Second Main Theorem, Green's description of the defect group in terms of vertices [4, 6] and a module description of the Brauer correspondence [1]. Our purpose here is to show how much of block theory can be done by statring with module-theoretic concepts; our contribution is the methods and not any new results, though we do hope these ideas can be applied elsewhere. We begin by defining defect groups in terms of vertices and so use Green's theorem to motivate the definition. This enables us to show quickly the fact that defect groups are Sylow intersections [4, 6] and can be characterized in terms of the vertices of the indecomposable modules in the block. We then define the Brauer correspondence in terms of modules and then prove part of the First Main Theorem, Nagao's theorem in full and results on blocks and normal subgroups.