The Torsion Theory at a Prime Ideal of a Right Noetherian Ring
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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. -
On the Cohen-Macaulay Modules of Graded Subrings
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 357, Number 2, Pages 735{756 S 0002-9947(04)03562-7 Article electronically published on April 27, 2004 ON THE COHEN-MACAULAY MODULES OF GRADED SUBRINGS DOUGLAS HANES Abstract. We give several characterizations for the linearity property for a maximal Cohen-Macaulay module over a local or graded ring, as well as proofs of existence in some new cases. In particular, we prove that the existence of such modules is preserved when taking Segre products, as well as when passing to Veronese subrings in low dimensions. The former result even yields new results on the existence of finitely generated maximal Cohen-Macaulay modules over non-Cohen-Macaulay rings. 1. Introduction and definitions The notion of a linear maximal Cohen-Macaulay module over a local ring (R; m) was introduced by Ulrich [17], who gave a simple characterization of the Gorenstein property for a Cohen-Macaulay local ring possessing a maximal Cohen-Macaulay module which is sufficiently close to being linear, as defined below. A maximal Cohen-Macaulay module (abbreviated MCM module)overR is one whose depth is equal to the Krull dimension of the ring R. All modules to be considered in this paper are assumed to be finitely generated. The minimal number of generators of an R-module M, which is equal to the (R=m)-vector space dimension of M=mM, will be denoted throughout by ν(M). In general, the length of a finitely generated Artinian module M is denoted by `(M)(orby`R(M), if it is necessary to specify the ring acting on M). -
On Prime Ideals, the Prime Radical and M-Systems Prabhjot Kaur Asst
Volume-9 • Number-1 Jan -June 2017 pp. 9-13 available online at www.csjournalss.com A UGC Recommended Journal http://ugc.ac.in/journallist vide letter Dated: 28/03/2017 On Prime Ideals, the Prime Radical and M-Systems Prabhjot Kaur Asst. Prof. D.A.V. College (Lahore), Ambala City Abstract: A group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and satisfy certain axioms while a ring is an algebraic structure with two binary operations namely addition and multiplication. I have tried to discuss prime ideals, prime radical and m- system in this paper. Keywords: Prime ideals, semi-prime ideals, m-system, n-system. 1. INTRODUCTION In this paper, I have tried to explain the concept of prime ideals in an arbitrary ring, Radical of a ring and few properties of m-system. Besides these some theorems and lemma have been raised such as “If A is an ideal in ring R then B(A) coincide with intersection of all prime ideals in R which contain A”. Also some theorems and lemmas based on m-system and n-system have been established. 1.1 Prime Integer: An integer p is said to be prime integer if it has following property that if a and b are integers such that ab is divisible by p then a is divisible by p or b is divisible by p. 1.2 Prime Ideal [1]: An ideal P in ring R is said to be a prime ideal if and only if it has the following property: If A and B are ideals in R such that AB P, then A P or B P 1.3 m-system: A set M of elements of a ring R is said to be an m-system if and only if it has following property: If a, b M, these exists x R such that axb M 1.4 Theorem: If P is an ideal in ring R, then following conditions are equivalent: (i) P is a prime ideal (ii) If a, b R such that aRb P then a P or b P (iii) If (a) and (b) are principal ideals in R such that (a) (b) P then a P or b P. -
Modules and Lie Semialgebras Over Semirings with a Negation Map 3
MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP GUY BLACHAR Abstract. In this article, we present the basic definitions of modules and Lie semialgebras over semirings with a negation map. Our main example of a semiring with a negation map is ELT algebras, and some of the results in this article are formulated and proved only in the ELT theory. When dealing with modules, we focus on linearly independent sets and spanning sets. We define a notion of lifting a module with a negation map, similarly to the tropicalization process, and use it to prove several theorems about semirings with a negation map which possess a lift. In the context of Lie semialgebras over semirings with a negation map, we first give basic definitions, and provide parallel constructions to the classical Lie algebras. We prove an ELT version of Cartan’s criterion for semisimplicity, and provide a counterexample for the naive version of the PBW Theorem. Contents Page 0. Introduction 2 0.1. Semirings with a Negation Map 2 0.2. Modules Over Semirings with a Negation Map 3 0.3. Supertropical Algebras 4 0.4. Exploded Layered Tropical Algebras 4 1. Modules over Semirings with a Negation Map 5 1.1. The Surpassing Relation for Modules 6 1.2. Basic Definitions for Modules 7 1.3. -morphisms 9 1.4. Lifting a Module Over a Semiring with a Negation Map 10 1.5. Linearly Independent Sets 13 1.6. d-bases and s-bases 14 1.7. Free Modules over Semirings with a Negation Map 18 2. -
Irreducible Representations of Finite Monoids
U.U.D.M. Project Report 2019:11 Irreducible representations of finite monoids Christoffer Hindlycke Examensarbete i matematik, 30 hp Handledare: Volodymyr Mazorchuk Examinator: Denis Gaidashev Mars 2019 Department of Mathematics Uppsala University Irreducible representations of finite monoids Christoffer Hindlycke Contents Introduction 2 Theory 3 Finite monoids and their structure . .3 Introductory notions . .3 Cyclic semigroups . .6 Green’s relations . .7 von Neumann regularity . 10 The theory of an idempotent . 11 The five functors Inde, Coinde, Rese,Te and Ne ..................... 11 Idempotents and simple modules . 14 Irreducible representations of a finite monoid . 17 Monoid algebras . 17 Clifford-Munn-Ponizovski˘ıtheory . 20 Application 24 The symmetric inverse monoid . 24 Calculating the irreducible representations of I3 ........................ 25 Appendix: Prerequisite theory 37 Basic definitions . 37 Finite dimensional algebras . 41 Semisimple modules and algebras . 41 Indecomposable modules . 42 An introduction to idempotents . 42 1 Irreducible representations of finite monoids Christoffer Hindlycke Introduction This paper is a literature study of the 2016 book Representation Theory of Finite Monoids by Benjamin Steinberg [3]. As this book contains too much interesting material for a simple master thesis, we have narrowed our attention to chapters 1, 4 and 5. This thesis is divided into three main parts: Theory, Application and Appendix. Within the Theory chapter, we (as the name might suggest) develop the necessary theory to assist with finding irreducible representations of finite monoids. Finite monoids and their structure gives elementary definitions as regards to finite monoids, and expands on the basic theory of their structure. This part corresponds to chapter 1 in [3]. The theory of an idempotent develops just enough theory regarding idempotents to enable us to state a key result, from which the principal result later follows almost immediately. -
On Prime Rings with Ascending Chain Condition on Annihilator Right Ideals and Nonzero Infective Right Ideals
Canad. Math. Bull. Vol. 14 (3), 1971 ON PRIME RINGS WITH ASCENDING CHAIN CONDITION ON ANNIHILATOR RIGHT IDEALS AND NONZERO INFECTIVE RIGHT IDEALS BY KWANGIL KOH AND A. C. MEWBORN If / is a right ideal of a ring R91 is said to be an annihilator right ideal provided that there is a subset S in R such that I={reR\sr = 0, VseS}. lis said to be injective if it is injective as a submodule of the right regular i£-module RR. The purpose of this note is to prove that a prime ring R (not necessarily with 1) which satisfies the ascending chain condition on annihilator right ideals is a simple ring with descending chain condition on one sided ideals if R contains a nonzero right ideal which is injective. LEMMA 1. Let M and T be right R-modules such that M is injective and T has zero singular submodule [4] and no nonzero injective submodule. Then Hom# (M, T)={0}. Proof. Suppose fe Hom^ (M, T) such that /=£ o. Let Kbe the kernel off. Then K is a proper submodule of M and there exists me M such that f(m)^0. Let (K:m)={reR\mreK}. Since the singular submodule of T is zero and f(m)(K:m)={0} the right ideal (K:m) has zero intersection with some nonzero right ideal / in R. Then ra/#{0} and K n mJ={0}. Let mJ be the injective hull of m J. Since M is injective, mJ is a submodule of M. mJC\ K={0} since m J has nonzero intersection with each submodule which has nonzero intersection with mJ (See [4, p. -
A Topology for Operator Modules Over W*-Algebras Bojan Magajna
Journal of Functional AnalysisFU3203 journal of functional analysis 154, 1741 (1998) article no. FU973203 A Topology for Operator Modules over W*-Algebras Bojan Magajna Department of Mathematics, University of Ljubljana, Jadranska 19, Ljubljana 1000, Slovenia E-mail: Bojan.MagajnaÄuni-lj.si Received July 23, 1996; revised February 11, 1997; accepted August 18, 1997 dedicated to professor ivan vidav in honor of his eightieth birthday Given a von Neumann algebra R on a Hilbert space H, the so-called R-topology is introduced into B(H), which is weaker than the norm and stronger than the COREultrastrong operator topology. A right R-submodule X of B(H) is closed in the Metadata, citation and similar papers at core.ac.uk Provided byR Elsevier-topology - Publisher if and Connector only if for each b #B(H) the right ideal, consisting of all a # R such that ba # X, is weak* closed in R. Equivalently, X is closed in the R-topology if and only if for each b #B(H) and each orthogonal family of projections ei in R with the sum 1 the condition bei # X for all i implies that b # X. 1998 Academic Press 1. INTRODUCTION Given a C*-algebra R on a Hilbert space H, a concrete operator right R-module is a subspace X of B(H) (the algebra of all bounded linear operators on H) such that XRX. Such modules can be characterized abstractly as L -matricially normed spaces in the sense of Ruan [21], [11] which are equipped with a completely contractive R-module multi- plication (see [6] and [9]). -
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]. -
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. -
D-Extending Modules
Hacettepe Journal of Hacet. J. Math. Stat. Volume 49 (3) (2020), 914 – 920 Mathematics & Statistics DOI : 10.15672/hujms.460241 Research Article D-extending modules Yılmaz Durğun Department of Mathematics, Çukurova University, Adana, Turkey Abstract A submodule N of a module M is called d-closed if M/N has a zero socle. D-closed submodules are similar concept to s-closed submodules, which are defined through non- singular modules by Goodearl. In this article we deal with modules with the property that all d-closed submodules are direct summands (respectively, closed, pure). The structure of a ring over which d-closed submodules of every module are direct summand (respectively, closed, pure) is studied. Mathematics Subject Classification (2010). 16D10, 16D40 Keywords. D-extending modules, d-closed submodules, semiartinian modules 1. Introduction Throughout we shall assume that all rings are associative with identity and all modules are unitary right modules. Let R be a ring and let M be an R-module. A (proper) submodule N of M will be denoted by (N M) N ≤ M. By E(M), Rad(M) and Soc(M) we shall denote the injective hull, the Jacobson radical, and the socle of M as usual. For undefined notions used in the text, we refer the reader to [2,11]. A submodule N of a module M is essential (or large) in M, denoted N ¢M, if for every 0 ≠ K ≤ M, we have N ∩ K ≠ 0; and N is said to be closed in M if N has no proper essential extension in M. We also say in this case that N is a closed submodule. -
Local Cohomology and Base Change
LOCAL COHOMOLOGY AND BASE CHANGE KAREN E. SMITH f Abstract. Let X −→ S be a morphism of Noetherian schemes, with S reduced. For any closed subscheme Z of X finite over S, let j denote the open immersion X \ Z ֒→ X. Then for any coherent sheaf F on X \ Z and any index r ≥ 1, the r sheaf f∗(R j∗F) is generically free on S and commutes with base change. We prove this by proving a related statement about local cohomology: Let R be Noetherian algebra over a Noetherian domain A, and let I ⊂ R be an ideal such that R/I is finitely generated as an A-module. Let M be a finitely generated R-module. Then r there exists a non-zero g ∈ A such that the local cohomology modules HI (M)⊗A Ag are free over Ag and for any ring map A → L factoring through Ag, we have r ∼ r HI (M) ⊗A L = HI⊗AL(M ⊗A L) for all r. 1. Introduction In his work on maps between local Picard groups, Koll´ar was led to investigate the behavior of certain cohomological functors under base change [Kol]. The following theorem directly answers a question he had posed: f Theorem 1.1. Let X → S be a morphism of Noetherian schemes, with S reduced. Suppose that Z ⊂ X is closed subscheme finite over S, and let j denote the open embedding of its complement U. Then for any coherent sheaf F on U, the sheaves r f∗(R j∗F) are generically free and commute with base change for all r ≥ 1. -
Notes on FP-Projective Modules and FP-Injective Modules
February 17, 2006 17:49 Proceedings Trim Size: 9in x 6in mao-ding NOTES ON FP -PROJECTIVE MODULES AND FP -INJECTIVE MODULES LIXIN MAO Department of Mathematics, Nanjing Institute of Technology Nanjing 210013, P.R. China Department of Mathematics, Nanjing University Nanjing 210093, P.R. China E-mail: [email protected] NANQING DING Department of Mathematics, Nanjing University Nanjing 210093, P.R. China E-mail: [email protected] In this paper, we study the FP -projective dimension under changes of rings, es- pecially under (almost) excellent extensions of rings. Some descriptions of FP - injective envelopes are also given. 1. Introduction Throughout this paper, all rings are associative with identity and all mod- ules are unitary. We write MR (RM) to indicate a right (left) R-module, and freely use the terminology and notations of [1, 4, 9]. 1 A right R-module M is called FP -injective [11] if ExtR(N; M) = 0 for all ¯nitely presented right R-modules N. The concepts of FP -projective dimensions of modules and rings were introduced and studied in [5]. For a right R-module M, the FP -projective dimension fpdR(M) of M is de¯ned to be the smallest integer n ¸ 0 such n+1 that ExtR (M; N) = 0 for any FP -injective right R-module N. If no such n exists, set fpdR(M) = 1. M is called FP -projective if fpdR(M) = 0. We note that the concept of FP -projective modules coincides with that of ¯nitely covered modules introduced by J. Trlifaj (see [12, De¯nition 3.3 and Theorem 3.4]).