On Commutative Ekfn-Ring with Ascending Chain Condition on Annihilators
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A Representation Theory for Noetherian Rings
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 39, 100-130 (1976) A Representation Theory for Noetherian Rings ROBERT GORDON* AND EDWARD L. GREEN+ Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122 Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania I9104 Communicated by A. W. Goldie Received October 28, 1974 The representation theory of artinian rings has long been studied, and seldommore intensely than in the past few years. However, with the exception of integral representationsof finite groups, there is no representation theory for noetherian rings even when they are commutative. Our aim in this paper is to develop such a theory. We attempt to pattern it after the known represen- tation theory of artinian rings. For this, we require a notion of indecom- posability different from the usual one; and we say that a module is strongly indecomposableif it is noetherian, and no factor by a nontrivial direct sum of submoduleshas lower Krull dimension than the module. Thus strongly indecomposablemodules are indecomposableand, by way of example, any uniform noetherian module is strongly indecomposable. Our first result states that some factor of a noetherian module by a direct sum of strongly indecomposablesubmodules has dimension less than the dimension of the module. This decomposition refines indecomposable noetherian modules, but we can say little about it without restricting the underlying ring. Thus we study, in Section 1, or-indecomposablemodules; that is, strongly indecomposablea-dimensional modules that have no nonzero submodule of dimension < cz. The resemblanceof these to indecomposable modules of finite length is analogousto that of a-critical modules to simple ones (see [8]). -
The Structure of Certain Artinian Rings with Zero Singular Ideal
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 8, 156-164 (1968) The Structure of Certain Artinian Rings with Zero Singular Ideal R. R. COLBY AND EDGAR A. RUTTER, JR. The University of Kansas, Lawrence, Kansas 66044 Communicated by A. W. Goldie Received March 4, 1967 INTRODUCTION Let R be a ring with identity and M and N (left) R-modules with MC N. Then M is said to be essential in N if M intersects every nonzero submodule of N nontrivially. A left ideal I of R is called an essential left ideal if I is essential in R. We denote the singular submodule of N by Z(N); 2(N)={n~NIhz=(O)f or some essential left ideal I of R}. We shall assume throughout this paper that Z(R) = 0. In Section 1 we outline the construction of Johnson’s ring of quotients [3] and some of its properties and develop certain other preliminary results. In Section 2 we determine the structure of Artinian rings with zero singular ideal having the property that each principal indecomposable left ideal contains a unique minimal left ideal. In particular we show that if R is such a ring then R is a certain distinguished subring of a complete blocked triangular matrix ring over a division subring of R. We also consider the case where R is a left generalized uniserial ring and generalize results of A. W. Goldie [2] and I. Murase [.5& Finally we apply the methods to obtain the structure of finite-dimensional algebras with zero singular ideal. -
Lie Algebras and Representation Theory Course Description
ALGEBRAIC STRUCTURES: A SECOND YEAR ALGEBRA SEQUENCE JULIA PEVTSOVA AND PAUL SMITH 1. 1st quarter: Lie algebras and Representation theory Course description. This course will cover the foundations of semi-simple Lie algebras, and their representation theory. It will also lay foundations for any further study of representation theory, Lie theory, and many other related topics. We shall discuss the structure of concrete examples of classical Lie algebras such as gln, sln, spn and son. This will be followed by the general theory of universal enveloping algebras and PBW theorem. The abstract theory of root systems and weight lattices that we shall develop will allow us to classify semi-simple Lie algebras. The second part of the course will be devoted to representations of Lie algebras and the theory of weights. Most of the theory will be developed over the complex numbers. Additional topics to discuss, time permitting, will be real Lie algebras and Lie algebra coho- mology. Topics at a glance. (1) Definitions and examples (2) Universal enveloping algebra and PBW theorem (3) Solvable and nilpotent Lie algebras (4) Root systems (5) Classification of semi-simple Lie algebras (6) Serre relations (7) Representation theory: (a) Highest weight modules (b) Finite dimensional irreducible modules, Dominant weights (c) Weyl character formula (8) Chevalley basis Reference. J. Humphreys, “Introduction to Lie algebras and Representation the- ory”. 1 2 JULIAPEVTSOVAANDPAULSMITH 2. 2nd quarter: Non-commutative Algebras Course description. This course will cover the foundations of finite- and infinite- dimensional associative algebras. The basic finite-dimensional examples will be the group algebra of a finite group, and the path algebra of appropriate quivers with relations. -
Artinian Subrings of a Commutative Ring
transactions of the american mathematical society Volume 336, Number 1, March 1993 ARTINIANSUBRINGS OF A COMMUTATIVERING ROBERT GILMER AND WILLIAM HEINZER Abstract. Given a commutative ring R, we investigate the structure of the set of Artinian subrings of R . We also consider the family of zero-dimensional subrings of R. Necessary and sufficient conditions are given in order that every zero-dimensional subring of a ring be Artinian. We also consider closure properties of the set of Artinian subrings of a ring with respect to intersection or finite intersection, and the condition that the set of Artinian subrings of a ring forms a directed family. 1. Introduction Suppose R is a commutative ring with unity element. We consider here various properties of the family sf of Artinian subrings of R and the family Z of zero-dimensional subrings of R . We remark that the inclusion s? ç Z may be proper, even if R is Noetherian; for example, Corollary 3.4 implies that if K is an infinite field of positive characteristic, then the local principal ideal ring K[X]/(X2) contains a zero-dimensional subring that is not Artinian. Of course, if every subring of R were Noetherian, the families sf and Z would be identical. Thus one source of motivation for this work comes from papers such as [Gi, Wi, W2, GHi, GH3] that deal with Noetherian and zero- dimensional pairs of rings, hereditarily Noetherian rings, and hereditarily zero- dimensional rings. Another source of motivation is related to our work in [GH3], where we considered several problems concerning a direct product of zero-dimensional rings. -
Integral Domains, Modules and Algebraic Integers Section 3 Hilary Term 2014
Module MA3412: Integral Domains, Modules and Algebraic Integers Section 3 Hilary Term 2014 D. R. Wilkins Copyright c David R. Wilkins 1997{2014 Contents 3 Noetherian Rings and Modules 49 3.1 Modules over a Unital Commutative Ring . 49 3.2 Noetherian Modules . 50 3.3 Noetherian Rings and Hilbert's Basis Theorem . 53 i 3 Noetherian Rings and Modules 3.1 Modules over a Unital Commutative Ring Definition Let R be a unital commutative ring. A set M is said to be a module over R (or R-module) if (i) given any x; y 2 M and r 2 R, there are well-defined elements x + y and rx of M, (ii) M is an Abelian group with respect to the operation + of addition, (iii) the identities r(x + y) = rx + ry; (r + s)x = rx + sx; (rs)x = r(sx); 1x = x are satisfied for all x; y 2 M and r; s 2 R. Example If K is a field, then a K-module is by definition a vector space over K. Example Let (M; +) be an Abelian group, and let x 2 M. If n is a positive integer then we define nx to be the sum x + x + ··· + x of n copies of x. If n is a negative integer then we define nx = −(jnjx), and we define 0x = 0. This enables us to regard any Abelian group as a module over the ring Z of integers. Conversely, any module over Z is also an Abelian group. Example Any unital commutative ring can be regarded as a module over itself in the obvious fashion. -
0.2 Structure Theory
18 d < N, then by Theorem 0.1.27 any element of V N can be expressed as a linear m m combination of elements of the form w1w2 w3 with length(w1w2 w3) ≤ N. By the m Cayley-Hamilton theorem w2 can be expressed as an F -linear combination of smaller n N−1 N N−1 powers of w2 and hence w1w2 w3 is in fact in V . It follows that V = V , and so V n+1 = V n for all n ≥ N. 0.2 Structure theory 0.2.1 Structure theory for Artinian rings To understand affine algebras of GK dimension zero, it is necessary to introduce the concept of an Artinian ring. Definition 0.2.1 A ring R is said to be left Artinian (respectively right Artinian), if R satisfies the descending chain condition on left (resp. right) ideals; that is, for any chain of left (resp. right) ideals I1 ⊇ I2 ⊇ I3 ⊇ · · · there is some n such that In = In+1 = In+2 = ··· : A ring that is both left and right Artinian is said to be Artinian. A related concept is that of a Noetherian ring. Definition 0.2.2 A ring R is said to be left Noetherian (respectively right Noethe- rian) if R satisfies the ascending chain condition on left (resp. right) ideals. Just as in the Artinian case, we declare a ring to be Noetherian if it is both left and right Noetherian. An equivalent definition for a left Artinian ring is that every non-empty collection of left ideals has a minimal element (when ordered under inclusion). -
Lectures on Non-Commutative Rings
Lectures on Non-Commutative Rings by Frank W. Anderson Mathematics 681 University of Oregon Fall, 2002 This material is free. However, we retain the copyright. You may not charge to redistribute this material, in whole or part, without written permission from the author. Preface. This document is a somewhat extended record of the material covered in the Fall 2002 seminar Math 681 on non-commutative ring theory. This does not include material from the informal discussion of the representation theory of algebras that we had during the last couple of lectures. On the other hand this does include expanded versions of some items that were not covered explicitly in the lectures. The latter mostly deals with material that is prerequisite for the later topics and may very well have been covered in earlier courses. For the most part this is simply a cleaned up version of the notes that were prepared for the class during the term. In this we have attempted to correct all of the many mathematical errors, typos, and sloppy writing that we could nd or that have been pointed out to us. Experience has convinced us, though, that we have almost certainly not come close to catching all of the goofs. So we welcome any feedback from the readers on how this can be cleaned up even more. One aspect of these notes that you should understand is that a lot of the substantive material, particularly some of the technical stu, will be presented as exercises. Thus, to get the most from this you should probably read the statements of the exercises and at least think through what they are trying to address. -
Certain Artinian Rings Are Noetherian
Can. J. Math., Vol. XXIV, No. 4, 1972, pp. 553-556 CERTAIN ARTINIAN RINGS ARE NOETHERIAN ROBERT C. SHOCK 1. Introduction. Throughout this paper the word "ring" will mean an associative ring which need not have an identity element. There are Artinian rings which are not Noetherian, for example C(pco) with zero multiplication. These are the only such rings in that an Artinian ring R is Noetherian if and only if R contains no subgroups of type C(pœ) [1, p. 285]. However, a certain class of Artinian rings is Noetherian. A famous theorem of C. Hopkins states that an Artinian ring with an identity element is Noetherian [3, p. 69]. The proofs of these theorems involve the method of "factoring through the nilpotent Jacobson radical of the ring". In this paper we state necessary and sufficient conditions for an Artinian ring (and an Artinian module) to be Noetherian. Our proof avoids the concept of the Jacobson radical and depends primarily upon the concept of the length of a composition series. As a corollary we obtain the result of Hopkins. All conditions are on the right, that is, an Artinian ring means a right Artinian ring, etc. We say a module A is embeddable in a module B provided that A is isomorphic to some submodule of B. Our Theorem 3 states that an Artinian ring R is Noetherian if and only if a cyclic Noetherian submodule ofR is embeddable in some Noetherian factor module of R. An example in the paper shows that in an Artinian ring R which is also Noetherian, a cyclic submodule need not be isomorphic to some factor module of R. -
Artinian Modules and Modules of Which All Proper Submodules Are Finitely Generated
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 95, 201-216 (1985) Artinian Modules and Modules of Which All Proper Submodules Are Finitely Generated WILLIAM HEINZER Purdue University, West Lafayette, Indiana 47907 AND DAVID LANTZ Purdue University, West Lafayette, Indiana 47907, and Colgate University, Hamilton, New York 13346 Communicated by I. N. Herstein Received September 5, 1983 1. INTRODUCTION William D. Weakley, in [Wl], studied modules (unitary, over a com- mutative ring with unity) which are not finitely generated but all of whose proper submodules are finitely generated. Weakley called such modules “almost finitely generated” (a.f.g.). He noted that an a.f.g. module has prime annihilator and that, as a faithful module over a domain, it is either torsion or torsion-free. If torsion-free, it is isomorphic as a module to the domain’s quotient field. If torsion, it is Artinian, so Weakley was led to the tools of study of Artinian modules developed by Matlis [Ml, M23 and Vamos [V]. These tools include, for a quasilocal ring (R, M), the injective envelope E,(R/M) of the residue field R/M. The purpose of the present paper is to combine results of [Wl] and [GH2] to characterize the form of a.f.g. modules and the rings which admit such modules, and to describe the a.f.g. submodules of E,(R/M). In Section 2, we refine a result of [ Wl ] to conclude that, if R is a domain and (F’, P) is a discrete rank one valuation domain between R and its quotient field K, then K/V is an a.f.g. -
Artinian Rings and Modules
ARTINIAN RINGS AND MODULES Let R be a ring (not necessarily commutative) and let M be a left (or right) R-module. Then M Artinian means that every simple descending chain of submodules M1 ⊃ M2 ⊃ M3 ⊃ · · · stabilizes, that is, for some r and all n ≥ 0, Mr = Mr+n. Equivalently, every non-empty family of submodules of M contains members that are minimal in that family. We say that R is left (or right) Artinian if it is Artinian as a left (or right) module over itself. To some extent, arguments with Artinian modules are very similar to arguments with Noetherian modules, that is, modules in which every simple ascending chain stabilizes. For example: LEMMA 1. If 0 ! M 0 ! M ! M 00 ! 0 is a short exact sequence of R-modules, then M is Artinian [resp. Noetherian] if and only if both M 0;M 00 are Artinian [resp. Noetherian]. PROOF. For the `if' direction in both cases, consider a chain of submodules Mm 00 of M. Projecting to M , the image chain stabilizes, say from the image of Mr. 00 From this point, the kernels of Mm ! M ; m ≥ r; form a chain of submodules 0 of M that stabilizes. The 5-lemma then implies the Mm stabilize as well. If M satisfies a chain condition then obviously so does M 0. But also, given a chain in M 00, the inverse images in M form a chain. Since M ! M 00 is surjective, the fact that the chain in M stabilizes implies the original chain in M 00 stabilizes. -
Minimal Ideals and Some Applications Rodney Coleman, Laurent Zwald
Minimal ideals and some applications Rodney Coleman, Laurent Zwald To cite this version: Rodney Coleman, Laurent Zwald. Minimal ideals and some applications. 2020. hal-03040754 HAL Id: hal-03040754 https://hal.archives-ouvertes.fr/hal-03040754 Preprint submitted on 4 Dec 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Minimal ideals and some applications Rodney Coleman, Laurent Zwald December 4, 2020 Abstract In these notes we introduce minimal prime ideals and some of their applications. We prove Krull's principal ideal and height theorems and introduce and study the notion of a system of parameters of a local ring. In addition, we give a detailed proof of the formula for the dimension of a polynomial ring over a noetherian ring. Let R be a commutative ring and I a proper ideal in R. A prime ideal P is said to be a minimal prime ideal over I if it is minimal (with respect to inclusion) among all prime ideals containing I. A prime ideal is said to be minimal if it is minimal over the zero ideal (0). Minimal prime ideals are those of height 0. -
Lecture 4: Artinian Rings
Lecture 4 1. General facts Proposition 1.1. Let A be a commutative ring, and m a maximal ideal. Then TFAE: (1) A has only one maximal ideal (i.e., A is local); (2) A n m consists of units in A; (3) For all non-units a and b, a + b is a nonunit. Proof. For (1) implies (2), if u2 = m, then u is a unit. For (2) implies (3), take a and b to be nonunits. Then a; b 2 m. This implies a+b 2 m, so a + b is a nonunit. For (3) implies (1), take I to be the set of all nonunits. Then I is an ideal: we only need to show that if a is a nonunit, λa is a nonunit for all λ 2 A. Assume not, and take b such that λab = 1 which implies a is a unit, a contradiction. Also, I is maximal since otherwise we can find a proper ideal J ≤ A containing I. But any element of J is a nonunit, since J is proper, so J ⊆ I which is impossible. Finally since m is a proper ideal we have that m consists of nonunits, so m ⊆ I. They are both maximal ideals, so m = I. Corollary 1.2. If the set of all nonunits is an ideal in A, then A is local and this ideal is the maximal one. Definition 1.3. Let Jac (A) be the intersection of all maximal ideals. Proposition 1.4. x 2 Jac (A) if and only if 8 a 2 A 1 + ax is a unit.