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Bulletin of Mathematical Sciences

, Volume 2, Issue 2, pp 225–279 | Cite as

Direct-sum decompositions of modules with semilocal endomorphism rings

  • Alberto Facchini
Open Access
Article

Abstract

According to the classical Krull–Schmidt Theorem, any module of finite composition length decomposes as a direct sum of indecomposable modules in an essentially unique way, that is, unique up to isomorphism of the indecomposable summands and a permutation of the summands. Modules that do not have finite composition length can have completely different behaviors. In this survey, we consider in particular the case of the modules M R whose endomorphism ring E := End(M R ) is a semilocal ring, that is, E/J(E) is a semisimple artinian ring. For instance, modules of finite composition length have a semilocal endomorphism ring, but several other classes of modules also have a semilocal endomorphism ring, for example artinian modules, finite direct sums of uniserial modules, finitely generated modules over commutative semilocal rings, and finitely presented modules over arbitrary semilocal rings. Several interesting phenomena appear in these cases. For instance, modules with a semilocal endomorphism ring have very regular direct-sum decompositions into indecomposables, their direct summands can be described via lattices, and direct-sum decompositions into indecomposables (=uniserial submodules) of finite direct sums of uniserial modules are described via their monogeny classes and their epigeny classes up to two permutations of the factors.

Keywords

Direct-sum decomposition Module Semilocal ring Endomorphism ring Uniserial module Artinian module 

Mathematics Subject Classification

16D70 16D90 16L30 18E05 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

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© The Author(s) 2012

Authors and Affiliations

  1. 1.Dipartimento di Matematica Pura e ApplicataUniversità di PadovaPaduaItaly
  2. 2.Middle East Center of Algebra and its Applications (MECAA)King Abdulaziz UniversityJeddahSaudi Arabia

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