CONSTRUCTION OF MODULES WITH A PRESCRIBED DIRECT SUM DECOMPOSITION

Year 2014, Volume: 15 Issue: 15, 218 - 248, 01.06.2014

Dolors Herbera

https://doi.org/10.24330/ieja.266249

Cited By: 1

Abstract

We give some criteria for recognizing local rings that allow us to
show that indecomposable AB5∗ modules over commutative rings and couniform
modules over noetherian commutative rings have a local endomorphism
ring. We also develop some theory on methods to construct modules with
a prescribed direct-sum decomposition. As an application we realize an interesting
class of commutative monoids as monoids of direct summands of a
direct sum of a countable number of copies of a suitable artinian cyclic module,
showing that there may appear a rich supply of direct summands that are
not a direct sum of artinian modules. An important gadget for proving our
realization result is a variation of a method for realizing a given ring as the
endomorphism ring of a cyclic (artinian) module due to Armendariz, Fisher
and Snider.

Keywords

AB5∗-module, artinian module, semilocal ring, couniform module, category equivalence, monoid, direct-sum, pullback, pushout

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• Departament de Matem`atiques
• Universitat Aut`onoma de Barcelona
• E-08193 Bellaterra (Barcelona), Spain
• e-mail: [e-mail]