Big indecomposable modules and direct-sum relations (original) (raw)

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.

Indecomposable modules of large rank over Cohen-Macaulay local rings

Transactions of The American Mathematical Society, 2008

In 1911, E. Steinitz determined the structure of all finitely generated modules over Dedekind domains. This structure is so simple that one is tempted to try to generalize Steinitz's result to a larger class of commutative rings. Indeed, in a recent series of papers [KL1, KL2, KL3] L. Klingler and L. Levy presented a classification, up to isomorphism, of all finitely generated modules over a class of commutative rings they call "Dedekind-like". We recall that a commutative Noetherian local ring (R, m, k) is Dedekind-like [KL1, Definition 2.5] provided R is one-dimensional and reduced, the integral closure R is generated by at most 2 elements as an R-module, and m is the Jacobson radical of R. (In [KL2, (1.1.3)] a further requirement is imposed: If R/m is a field, then it is a separable extension of k. Klingler and Levy prove their classification theorem only under this additional hypothesis. In the present paper, however, we do not require that Dedekind-like rings satisfy this separability condition.) Although Dedekind-like rings are very close to their normalizations, their module structure is much more complicated than that of Dedekind domains. Klingler and Levy dash any hope of a further extension of their classification theorem by showing that, if R is not a homomorphic image of a Dedekind-like ring or a special type of Artinian ring which they call a Klein ring, then R must be "finite-length wild". This means, roughly speaking, that a classification of finite-length modules over R would yield, for some field k, a classification of finite-dimensional modules over every finite-dimensional k-algebra. The apparent hopelessness of obtaining such a classification makes any further generalizations of Steinitz's result unlikely. One of the peculiarities of Dedekind-like rings is that there is a bound on the torsion-free ranks of their indecomposable finitely generated modules; in fact, these torsion-free ranks are always bounded by two. Recently W. Hassler and R. Wiegand [HaW] constructed an indecomposable finitely generated module of torsion-free rank two over the cusp k[[X 2 , X 3 ]] (where k is an arbitrary field) and some related rings. The approach they used to build

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

A note on indecomposable modules

Rendiconti del Circolo Matematico di Palermo, 1988

In this note we study rings having only a finite number of non isomorphic uniform modules with non zero socle. It is proved that a commutative ring with this property is a direct sum of a finite ring and a ring of finite representation type. In the non commutative case we show that most P.I. rings having only a finite number of non isomorphic modules with non zero socle are in fact artinlan.

Direct sums of indecomposable modules

OSAKA JOURNAL OF MATHEMATICS

1. Introduction. This paper studies direct sums M=(B iζΞI M i of indecomposable modules. Specifically, we give a number of necessary and sufficient conditions for such a sum to be quasi-continuous or continuous. This question was settled in [6], in a very satisfactory way, in case the index set / is finite, or the ring is right-noetherian, but the general case dealt with here is much more complicated. Such sums M=@ i^I M i have been investigated in great detail, in a long series of papers since about 1970, by M. Harada and his collaborators, usually under the additional hypothesis that the M { have local endomorphism rings (so that the Krull-Schmidt-Azumaya Theorem applies). One of the central results is the following: Theorem 1 ([3], p. 22). For a module with a decomposition M=® ieI M h and with all endoίM,) local, the following statements are equivalent: (1) The decomposition is locally semi-T-nilpotent; (2) it complements direct summands (3) any local direct summand of M is a direct summand. (The relevent terms are defined later on in this section.

On the index of reducibility in Noetherian modules

Journal of Pure and Applied Algebra, 2015

Let M be a finitely generated module over a Noetherian ring R and N a submodule. The index of reducibility ir M (N) is the number of irreducible submodules that appear in an irredundant irreducible decomposition of N (this number is well defined by a classical result of Emmy Noether). Then the main results of this paper are: (1) ir M (N) = p∈Ass R (M/N) dim k(p) Soc(M/N) p ; (2) For an irredundant primary decomposition of N = Q 1 ∩ • • • ∩ Q n , where Q i is p i-primary, then ir M (N) = ir M (Q 1) + • • • + ir M (Q n) if and only if Q i is a p imaximal embedded component of N for all embedded associated prime ideals p i of N ; (3) For an ideal I of R there exists a polynomial Ir M,I (n) such that Ir M,I (n) = ir M (I n M) for n ≫ 0. Moreover, bight M (I) − 1 ≤ deg(Ir M,I (n)) ≤ ℓ M (I) − 1; (4) If (R, m) is local, M is Cohen-Macaulay if and only if there exist an integer l and a parameter ideal q of M contained in m l such that ir M (qM) = dim R/m Soc(H d m (M)), where d = dim M .