Absolutely indecomposable modules

A module is called absolutely indecomposable if it is directly indecomposable in every generic extension of the universe. We want to show the existence of large abelian groups that are absolutely indecomposable. This will follow from a more general result about -modules over a large class of commutative rings with endomorphism ring which remains the same when passing to a generic extension of the universe. It turns out that `large' in this context has a precise meaning, namely being smaller than the first -Erdos cardinal defined below. We will first apply a result on large rigid valuated trees with a similar property established by Shelah in 1982, and will prove the existence of related `-modules' (-modules with countably many distinguished submodules) and finally pass to -modules. The passage through -modules has the great advantage that the proofs become very transparent essentially using a few `linear algebra' arguments also accessible for graduate students. The result closes a gap of Eklof and Shelah (1999) and Eklof and Mekler (2002), provides a good starting point for Fuchs and Göbel, and gives a new construction of indecomposable modules in general using a counting argument.

     

不可分解模的参量化

本文的主要结果是给出了有限群的不可分解模的一种参量化, 它相对于Puig 和Thévenaz 给出的参量化更加自然. 同时我们说明了用这两种不同的方式给出的参量化是一致的.     

Free submodules of indecomposable modules

Let be an artin algebra which has only finitely many isomorphism classes of non-faithful indecomposable modules. We show that large indecomposable -modules contain large free submodules. Received: 12 April 2003     

A note on indecomposable modules

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 artinian.     

Bounds for indecomposable torsion-free modules

Let M be a finitely generated torsion-free module over a one-dimensional reduced Noetherian ring R with finitely generated normalization. The rank of M is the tuple of vector-space dimensions of M_P over each field R_P (R localized at P), where P ranges over the minimal prime ideals of R. We assume that there exists a bound N_R on the ranks of all indecomposable finitely generated torsion-free R-modules. For such rings, what bounds and ranks occur? Partial answers to this question have been given by a plethora of authors over the past forty years. In this article we provide a final answer by giving a concise list of the ranks of indecomposable modules for R a local ring with no condition on the characteristic. We conclude that if the rank of an indecomposable module M is (r,r,…,r), then r∈{1,2,3,4,6}, even when R is not local.     

Rank-sets of indecomposable modules over one-dimensional rings

Let R be a one-dimensional, reduced Noetherian ring with finite normalization, and suppose there exists a positive integer N_R such that, for every indecomposable finitely generated torsion-free R-module M and every minimal prime ideal P of R, the dimension of M_P, as a vector space over the localization R_P (a field), is less than or equal to N_R. For a finitely generated torsion-free R-module M, we call the set of all such vector-space dimensions the rank-set of M. What subsets of the integers arise as rank-sets of indecomposable finitely generated torsion-free R-modules? In this article, we give more information on rank-sets of indecomposable modules, to supplement previous work concerning this question. In particular we provide examples having as rank-sets those intervals of consecutive integers that are not ruled out by an earlier article of Arnavut, Luckas and Wiegand. We also show that certain non-consecutive rank-sets never arise.     

On the composition factors of indecomposable modules in almost cyclic coherent Auslander-Reiten components

We prove that the indecomposable modules without selfextensions in generalized standard almost cyclic coherent Auslander-Reiten components without external short paths of artin algebra are uniquely determined by their composition factors. Moreover, we prove that there is a common bound on the numbers of indecomposable modules with the same composition factors lying in a generalized standard almost cyclic coherent Auslander-Reiten component without external short paths of artin algebra.     

On indecomposable modules over the Virasoro algebra

It is proved that an indecomposable Harish- Chandra module over the Virasoro algebra must be (i) a uniformly bounded module, or (ii) a module in Category , or (iii) a module in Category , or ( iv) a module which contains the trivial module as one of its composition factors.     

On the structure of indecomposable injective modules 1

For a ring R, the following two conditions are equivalent:.

(1) If E is an indecomposable injective right R-module, then End E_R is a field (not necesarily commutative).

(2) Every co-irreducible rigtht ideal is critical.

Since (2) has been characterized ideal-theoretically, this amounts to an ideal-theoretical characterization of (1). These rings come up to the study of (QI) rings in which every quasi-injective module is injective.     

Selfinjective algebras without short cycles of indecomposable modules

We describe the structure of finite dimensional selfinjective algebras over an arbitrary field without short cycles of indecomposable modules.     

Rigidity degrees of indecomposable modules over representation-finite self-injective algebras

The rigidity degree of a generator-cogenerator determines the dominant dimension of its endomorphism algebra, and is closely related to a recently introduced homological dimension — rigidity dimension. In this paper, we give explicit formulae for the rigidity degrees of all indecomposable modules over representation-finite self-injective algebras by developing combinatorial methods from the Euclidean algorithm. As an application, the rigidity dimensions of some algebras of types A and E are given.     

The trivial intersection problem for characters of principal indecomposable modules

For a prime p and a finite group G let Φ_p(G) denote the complex character associated to the projective indecomposable module in characteristic p with trivial head. Let Irr(Φ_p(G)) denote the set of irreducible characters occurring as constituents in Φ_p(G). We characterize all finite simple groups which satisfy Irr(Φ_p(G))∩Irr(Φ_q(G))={1_G} for all primes p≠q.