Notes on Commutative Algebra
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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. -
Ring Extension Problem, Shukla Cohomology and Ann-Category Theory the Relationship Between Ring Extension Problem in the General Case and Ann- Category Theory
RING EXTENSION PROBLEM, SHUKLA COHOMOLOGY AND ANN-CATEGORY THEORY Nguyen Tien Quang and Nguyen Thu Thuy November 4, 2018 Abstract. Every ring extension of A by R induces a pair of group ho- ∗ ∗ momorphisms L : R → EndZ(A)/L(A); R : R → EndZ(A)/R(A), preserving multiplication, satisfying some certain conditions. A such 4-tuple (R, A, L∗, R∗) is called a ring pre-extension. Each ring pre-extension induces a R-bimodule structure on bicenter KA of ring A, and induces an obstruction k, which is a 3-cocycle of Z-algebra R, with coefficients in R-bimodule KA in the sense of Shukla. Each obstruction k in this sense induces a structure of a regular Ann-category of type (R,KA). This result gives us the first application of Ann- category in extension problems of algebraic structures, as well as in cohomology theories. 1 Introduction Group extension problem has been presented with group of automorphisms Aut(G) and quotient group Aut(G)/In(G) by group of inner automorphisms. For ring extension problem, Mac Lane [1] has replaced the above groups by the ring of bimultiplications MA and the quotient ring PA of MA upon ring of in- ner bimultiplications. Besides, Maclane has replaced commutative, associative laws for addition in ring R by commutative-associative law (u + v) + (r + s)= arXiv:0706.0315v1 [math.CT] 3 Jun 2007 (u+r)+(v+s) and therefore proved that each obstruction of ring extension prob- lem is an element of 3-dimensional cohomology group in the sense of Maclane, and the number of solutions corresponds to 2-dimensional cohomology group of ring under a bijection. -
A Non-Slick Proof of the Jordan Hölder Theorem E.L. Lady This Proof Is An
A Non-slick Proof of the Jordan H¨older Theorem E.L. Lady This proof is an attempt to approximate the actual thinking process that one goes through in finding a proof before one realizes how simple the theorem really is. To start with, though, we motivate the Jordan-H¨older Theorem by starting with the idea of dimension for vector spaces. The concept of dimension for vector spaces has the following properties: (1) If V = {0} then dim V =0. (2) If W is a proper subspace of V ,thendimW<dim V . (3) If V =6 {0}, then there exists a one-dimensional subspace of V . (4) dim(S U ⊕ W )=dimU+dimW. W chain (5) If SI i is a of subspaces of a vector space, then dim Wi =sup{dim Wi}. Now we want to use properties (1) through (4) as a model for defining an analogue of dimension, which we will call length,forcertain modules over a ring. These modules will be the analogue of finite-dimensional vector spaces, and will have very similar properties. The axioms for length will be as follows. Axioms for Length. Let R beafixedringandletM and N denote R-modules. Whenever length M is defined, it is a cardinal number. (1) If M = {0},thenlengthM is defined and length M =0. (2) If length M is defined and N is a proper submodule of M ,thenlengthN is defined. Furthermore, if length M is finite then length N<length M . (3) For M =0,length6 M=1ifandonlyifM has no proper non-trivial submodules. -
Isomorphic Groups and Group Rings
Pacific Journal of Mathematics ISOMORPHIC GROUPS AND GROUP RINGS DONALD STEVEN PASSMAN Vol. 15, No. 2 October 1965 PACIFIC JOURNAL OF MATHEMATICS Vol. 15, No. 2, 1965 ISOMORPHIC GROUPS AND GROUP RINGS D. S. PASSMAN Let © be a finite group, £ a commutative ring with one and S[@] the group ring of © over S. If ξ> is a group with © = £ then clearly S[(S] = S[£>] where the latter is an S-iso- morphism. We study here the converse question: For which groups © and rings S does £[©] ^ S[ξ>] imply that © is iso- morphic to £)? We consider first the case where S = K is a field. It is known that if © is abelian then Q[@] = Q[ξ>] implies that © = §> where Q is the field of rational numbers. We show here that this result does not extend to all groups ©. In fact by a simple counting argument we exhibit a large set of noniso- morphic p-groups with isomorphic group algebras over all noncharacteristic p fields. Thus for groups in general the only fields if interest are those whose characteristic divides the order of the group. We now let S = R be the ring of integers in some finite algebraic extension of the rationale. We show here that the group ring R[@>] determines the set of normal subgroups of © along with many of the natural operations defined on this set. For example, under the assumption that © is nilpotent, we show that given normal subgroups 3Dΐ and 9ΐ, the group ring determines the commutator subgroup (3JI, 91). Finally we consider several special cases. In particular we show that if © is nilpotent of class 2 then R[(g\ = β[§] implies © = €>. -
The Pseudo-Radical of a Commutative Ring
Pacific Journal of Mathematics THE PSEUDO-RADICAL OF A COMMUTATIVE RING ROBERT WILLIAM GILMER,JR. Vol. 19, No. 2 June 1966 PACIFIC JOURNAL OF MATHEMATICS Vol. 19, No. 2, 1966 THE PSEUDO-RADICAL OF A COMMUTATIVE RING ROBERT W. GILMER, JR. If D is an integral domain with identity having quotient field K, the pseudo-radical of D is defined to be the inter- section of all nonzero prime ideals of D. Consideration of the pseudo-radical arises naturally in examining the relation between the statements "D has Jacobson radical zero" and "D[u] has Jacobson radical zero, where ueK". Theorem 4 proves that the first statement implies the second. As a corollary it follows that if M is a prime ideal of the polynomial ring R[X] over a commutative ring R and if P=MnR, then M is an intersection of maximal ideals of R[X] if P is an intersection of maximal ideals of R. Consequently, if R is a Hubert ring, R[X] is also a Hubert ring. The remainder of the paper is devoted to a study of domains having nonzero pseudo-radical. Goldman has defined in [6] the concept of a Hubert ring: the commutative ring R with identity is a Hubert ring if each proper prime ideal of R is an intersection of maximal ideals; here proper means an ideal different from R. The terminology is motivated by the observation that Hubert's Nullstellensatz may be interpreted as asserting that each proper prime ideal of the polynomial domain K[Xlf - —, Xn] for K a field, is an intersection of maximal ideals. -
Trivial Ring Extension of Suitable-Like Conditions and Some
An - Najah Univ. J. Res.(N. Sc.) Vol. 33(1), 2019 Trivial Ring Extension of Suitable-Like Conditions and some properties AmÌ'@ ªK.ð AîEAîD J.ð éJ¯A¾JÖÏ@ HA ®ÊjÊË éJ îE YJ.Ë@ éJ ®ÊmÌ'@ H@YK YÒJË@ Khalid Adarbeh* éK.P@Y« YËAg Department of Mathematics, Faculty of Sciences, An-Najah National University, Nablus, Palestine * Corresponding Author: [email protected] Received: (6/9/2018), Accepted: (6/12/2018) Abstract We investigate the transfer of the notion of suitable rings along with re- lated concepts, such as potent and semipotent rings, in the general context of the trivial ring extension, then we put these results in use to enrich the literature with new illustrative and counter examples subject to these ring- theoretic notions. Also we discuss some basic properties of the mentioned notions. Keywords: Trivial ring extension, idealization, clean ring, potent ring, semipotent ring, von Neumann regular ring, suitable rings. jÊÓ Ì « ªK. ð éJ¯A¾ÖÏ@ HA®Êm '@ Ðñê®Ó ÈA®JK@ éJ ÊÔ PñÖÏ@@Yë ú ¯ Ij.K Ì Ì áÓ éÓAªË@ éËAm '@ ú ¯ éK ñ®Ë@ éJ. ð éK ñ®Ë@ HA®Êm '@ ú ×ñê®Ò» é¢J.KQÖÏ@ Õæ ëA®ÖÏ@ YK ðQ K ú¯ èYK Ym.Ì'@ l.'AJJË@ ÐYjJ Õç' áÓð , éJ îE YJ.Ë@ éJ ®ÊmÌ'@ H@YK YÒJË@ ªK . ¯AJ K AÒ» . éðPYÖÏ@ ÕæëA®ÒÊË © m' éJ j ñ K éÊJÓAK . HA«ñJ .¢ÖÏ@ .Õæ ëA®ÖÏ@ èYêË éJ ®ÊmÌ'@ AmÌ'@ , é®J ¢ JË@ é®ÊmÌ'@ , éJ ËAJÖ ÏAK. ÉÒªË@ , éJ îE YJ.Ë@ éJ ®ÊmÌ'@ H@YK YÒJË@ : éJ kAJ®ÖÏ@ HAÒʾË@ . -
Commutative Algebra
Commutative Algebra Andrew Kobin Spring 2016 / 2019 Contents Contents Contents 1 Preliminaries 1 1.1 Radicals . .1 1.2 Nakayama's Lemma and Consequences . .4 1.3 Localization . .5 1.4 Transcendence Degree . 10 2 Integral Dependence 14 2.1 Integral Extensions of Rings . 14 2.2 Integrality and Field Extensions . 18 2.3 Integrality, Ideals and Localization . 21 2.4 Normalization . 28 2.5 Valuation Rings . 32 2.6 Dimension and Transcendence Degree . 33 3 Noetherian and Artinian Rings 37 3.1 Ascending and Descending Chains . 37 3.2 Composition Series . 40 3.3 Noetherian Rings . 42 3.4 Primary Decomposition . 46 3.5 Artinian Rings . 53 3.6 Associated Primes . 56 4 Discrete Valuations and Dedekind Domains 60 4.1 Discrete Valuation Rings . 60 4.2 Dedekind Domains . 64 4.3 Fractional and Invertible Ideals . 65 4.4 The Class Group . 70 4.5 Dedekind Domains in Extensions . 72 5 Completion and Filtration 76 5.1 Topological Abelian Groups and Completion . 76 5.2 Inverse Limits . 78 5.3 Topological Rings and Module Filtrations . 82 5.4 Graded Rings and Modules . 84 6 Dimension Theory 89 6.1 Hilbert Functions . 89 6.2 Local Noetherian Rings . 94 6.3 Complete Local Rings . 98 7 Singularities 106 7.1 Derived Functors . 106 7.2 Regular Sequences and the Koszul Complex . 109 7.3 Projective Dimension . 114 i Contents Contents 7.4 Depth and Cohen-Macauley Rings . 118 7.5 Gorenstein Rings . 127 8 Algebraic Geometry 133 8.1 Affine Algebraic Varieties . 133 8.2 Morphisms of Affine Varieties . 142 8.3 Sheaves of Functions . -
Proquest Dissertations
University of Alberta Descent Constructions for Central Extensions of Infinite Dimensional Lie Algebras by Jie Sun A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Department of Mathematical and Statistical Sciences Edmonton, Alberta Spring 2009 Library and Archives Bibliotheque et 1*1 Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington OttawaONK1A0N4 OttawaONK1A0N4 Canada Canada Your file Votre rSterence ISBN: 978-0-494-55620-7 Our file Notre rGterence ISBN: 978-0-494-55620-7 NOTICE: AVIS: The author has granted a non L'auteur a accorde une licence non exclusive exclusive license allowing Library and permettant a la Bibliotheque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par I'lnternet, preter, telecommunication or on the Internet, distribuer et vendre des theses partout dans le loan, distribute and sell theses monde, a des fins commerciales ou autres, sur worldwide, for commercial or non support microforme, papier, electronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in this et des droits moraux qui protege cette these. Ni thesis. Neither the thesis nor la these ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent etre im primes ou autrement printed or otherwise reproduced reproduits sans son autorisation. -
[Math.AC] 3 Apr 2006
ON THE GROWTH OF THE BETTI SEQUENCE OF THE CANONICAL MODULE DAVID A. JORGENSEN AND GRAHAM J. LEUSCHKE Abstract. We study the growth of the Betti sequence of the canonical module of a Cohen–Macaulay local ring. It is an open question whether this sequence grows exponentially whenever the ring is not Gorenstein. We answer the ques- tion of exponential growth affirmatively for a large class of rings, and prove that the growth is in general not extremal. As an application of growth, we give criteria for a Cohen–Macaulay ring possessing a canonical module to be Gorenstein. Introduction A canonical module ωR for a Cohen–Macaulay local ring R is a maximal Cohen– Macaulay module having finite injective dimension and such that the natural ho- momorphism R HomR(ωR,ωR) is an isomorphism. If such a module exists, then it is unique−→ up to isomorphism. The ring R is Gorenstein if and only if R itself is a canonical module, that is, if and only if ωR is free. Although the cohomological behavior of the canonical module, both in algebra and in geometry, is quite well understood, little is known about its homological aspects. In this note we study the growth of the Betti numbers—the ranks of the free modules occurring in a min- imal free resolution—of ωR over R. Specifically, we seek to answer the following question, a version of which we first heard from C. Huneke. Question. If R is not Gorenstein, must the Betti numbers of the canonical module grow exponentially? By exponential growth of a sequence bi we mean that there exist real numbers i i { } 1 <α<β such that α < bi < β for all i 0. -
Nil and Jacobson Radicals in Semigroup Graded Rings
Faculty of Science Departement of Mathematics Nil and Jacobson radicals in semigroup graded rings Master thesis submitted in partial fulfillment of the requirements for the degree of Master in Mathematics Carmen Mazijn Promotor: Prof. Dr. E. Jespers AUGUST 2015 Acknowledgements When we started our last year of the Master in Mathematics at VUB, none of us knew how many hours we would spend on the reading, understanding and writing of our thesis. This final product as conclusion of the master was at that point only an idea. The subject was chosen, the first papers were read and the first words were written down. And more words were written, more books were consulted, more questions were asked to our promoters. Writing a Master thesis is a journey. Even though next week everyone will have handed in there thesis, we don’t yet understand clearly where this journey took us, for the future is unknown. First of all I would like to thank professor Eric Jespers, for giving me the chance to grow as mathematician in the past years. With every semester the interest in Algebra and accuracy as mathematician grew. Thank you for the guidance through all the books and papers to make this a consistent dissertation. Secondly I would like to thank all my classmates and compa˜nerosde clase. For frowned faces when we didn’t get something in class, the laughter when we realized it was a ctually quite trivial or sometimes not even at all. For the late night calls and the interesting discussions. It was a pleasure. -
Extensions of a Ring by a Ring with a Bimodule Structure
EXTENSIONS OF A RING BY A RING WITH A BIMODULE STRUCTURE CARL W. KOHLS Abstract. A type of ring extension is considered that was intro- duced by J. Szendrei and generalizes many familiar examples, including the complex extension of the real field. We give a method for constructing a large class of examples of this type of extension, and show that for some rings all possible examples are obtained by this method. An abstract characterization of the extension is also given, among rings defined on the set product of two given rings. This paper is a sequel to [2 ], in which a class of examples was given of a type of ring extension defined in terms of two functions. Here we exhibit some other functions that may be used to construct such extensions, and show that in certain cases (in particular, when the first ring is an integral domain and the second is a commutative ring with identity), the functions must have a prescribed form. We also characterize this type of ring extension, which Szendrei defined di- rectly by the ring operations, in terms of the manner in which the two given rings are embedded in the extension. The reader is referred to [2] for background. Only the basic defini- tion is repeated here. Let A and B be rings. We define the ring A*B to be the direct sum of A and B as additive groups, with multiplication given by (1) (a,b)(c,d) = (ac+ {b,d},aad + b<rc+ bd), where a is a homomorphism from A onto a ring of permutable bimul- tiplications of B, and { •, • } is a biadditive function from BXB into A satisfying the equations (2) bo-{c,d) = o-[b.c)d, (3) {b,cd\ = {bc,d\, (4) {b,aac\ = {baa,c\, (5) \o-ab,c} = a{b,c}, and {&,c<ra} = \b,c\a, for all a£^4 and all b, c, dEB. -
Chains and Lengths of Modules
Chains and Lengths of Modules A mathematical essay by Wayne Aitken∗ Fall 2019y In this essay begins with the descending and ascending chain conditions for modules, and more generally for partially ordered sets. If a module satisfies both of these chain conditions then it has an invariant called the length. This essay will consider some properties of this length. The first version of this essay was written to provide background material for another essay focused on the Krull{Akizuki theorem and on Akizuki's other theorem that every Artinian ring is Noetherian.1 The current essay considers Noetherian and Artinian modules, the follow-up essay considers Noetherian and Artinian rings. I have attempted to give full and clear statements of the definitions and results, and give indications of any proof that is not straightforward. However, my philos- ophy is that, at this level of mathematics, straightforward proofs are best worked out by the reader. So some of the proofs may be quite terse or missing altogether. Whenever a proof is not given, this signals to the reader that they should work out the proof, and that the proof is straightforward. Often supplied proofs are sketches, but I have attempted to be detailed enough that the reader can supply the details without too much trouble. Even when a proof is provided, I encourage the reader to attempt a proof first before looking at the provided proof. Often the reader's proof will make more sense because it reflects their own viewpoint, and may even be more elegant. 1 Required background This document is written for readers with some basic familiarity with abstract algebra including some basic facts about rings (at least commutative rings), ideals, modules over such rings, quotients of modules, and module homomorphisms and isomorphisms.