A note on indecomposable modules (original) (raw)

Indecomposable modules over right pure semisimple rings

Monatshefte f�r Mathematik, 1988

The aim of this paper is to prove the following result. If A is a fight pure semisimple ring, then it satisfies one of the two following statements: (a) For any positive integer n, there are at most finitely many indecomposable right modules of length n; or (b) There is an infinite number of integers d such that, for each d, A has infinitely many indecomposable right modules of length d. The result is derived with the aid of ultraproduct-technique.

Rings with Indecomposable Right Modules Local

Taiwanese Journal of Mathematics, 2010

Every indecomposable module over a generalized uniserial ring is uniserial, hence local. This motivates one to study rings R satisfying the condition (*): R is a right artinian ring such that every finitely generated, indecomposable right R-module is local. The rings R satisfying (*) have been recently studied by Singh and Al-Bleahed (2004), they have proved some results giving the structure of local right R-modules. In this paper some more structure theorems for local right R-modules are proved. Examples given in this paper show that a rich class of rings satisfying condition (*) can be constructed. Using these results, it is proved that any ring R satisfying (*) is such that mod-R is of finite representation type. It follows from a theorem by Ringel and Tachikawa that any right R-module is a direct sum of local modules. If M is right module over a right artinian ring such that any finitely generated submodule of any homomorphic image of M is a direct sum of local modules, it is proved that it is a direct sum of local modules. This provides an alternative proof for that any right module over a right artinian ring R satisfying (*) is a direct sum of local modules.

One-Dimensional Rings of Finite Representation Type

Abelian Groups and Modules, 1995

Let R be a commutative, semilocai, Noetherian domain, not a field. We say that R has finite representation type provided R has, up to isomorphism, only finitely many indecomposable finitely generated torsion-free modules. A special case (0.6) of our main theorem states that R has finite representation type if and only if 1. R has Krull dimension 1; 2. The integral closure R of R in its quotient field can be generated by 3 elements as an R-module; and 3. The intersection of the maximal R-submodules of RI R is a cyclic R-module. While there is no uniform bound on the number of indecomposable finitely generated torsionfree R-modules (as R varies among semilocal domains of finite representation type), the rank of every indecomposable is 1, 2, 3, 4, 5, 6, 8, 9 or 12. Moreover, there exists a semilocal domain R of finite representation type which has indecomposables of each of these ranks.

On algebras of finite representation type

Transactions of The American Mathematical Society, 1969

Introduction. Since D. G. Higman proved that bounded representation type and finite representation type are equivalent for group algebras at prime characteristic, there has been a renewed interest in the Brauer-Thrall conjecture that bounded representation type implies finite representation type for arbitrary algebras. The main purpose of this paper is to present a new approach to this conjecture by showing the relevance (when the base field is algebraically closed) of questions concerning the structure of indecomposable modules of certain special types, namely, the stable (every maximal submodule is indecomposable), the costable (having the dual property), and the stable-costable (having both properties) indecomposable modules. The main tools are the Sandwich Lemma (1.2) which is proved using an old observation of É. Goursat, an observation of A. Heller, C. W. Curtis, and D. Zelinsky concerning quasifrobenius (QF) rings (Proposition 2.1), and a general interlacing technique similar to methods used by Jans, Tachikawa, and Colby for building up large indecomposable modules of finite length which has validity in any abelian category (Theorem 3.1).

Big indecomposable modules and direct-sum relations

Illinois Journal of Mathematics - ILL J MATH, 2007

A commutative Noetherian local ring (R,m)(R,\m)(R,m) is said to be \emph{Dedekind-like} provided RRR has Krull-dimension one, RRR has no non-zero nilpotent elements, the integral closure overlineR\overline RoverlineR of RRR is generated by two elements as an RRR-module, and m\mm is the Jacobson radical of overlineR\overline RoverlineR. A classification theorem due to Klingler and Levy implies that if MMM is a finitely generated indecomposable module over a Dedekind-like ring, then, for each minimal prime ideal PPP of RRR, the vector space MPM_PMP has dimension 0,10, 10,1 or 222 over the field RPR_PRP. The main theorem in the present paper states that if RRR (commutative, Noetherian and local) has non-zero Krull dimension and is not a homomorphic image of a Dedekind-like ring, then there are indecomposable modules that are free of any prescribed rank at each minimal prime ideal.

Modules with Finitely Many Submodules

Algebra Colloquium, 2016

In this paper, we study modules having only finitely many submodules over any ring which is not necessarily commutative. We try to understand how such a module decomposes as a direct sum. We justify that any module V having only finitely many submodules over any ring A is an extension of a cyclic A-module by a finite A-module. Under some assumptions on A, such as commutativity of A, we prove that an A-module V has finitely many submodules if and only if V can be written as a direct sum of a cyclic A-module having only finitely many A-submodules and a finite A-module.

Constructing Big Indecomposable Modules

2013

Abstract. Let R be local Noetherian ring of depth at least two. We prove that there are indecomposable R-modules which are free on the punctured spectrum of constant, arbitrarily large, rank. 1. introduction A fruitful approach to study a commutative ring is to understand the category of its finitely generated modules, and in particular the indecomposable objects of such a category. Over zero dimensional rings it is feasible to understand all the

Large indecomposable modules over local rings

Journal of Algebra, 2006

For commutative, Noetherian, local ring R of dimension one, we show that, if R is not a homomorphic image of a Dedekind-like ring, then R has indecomposable finitely generated modules that are free of arbitrary rank at each minimal prime. For Cohen-Macaulay ring R, this theorem was proved in [W.