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Characterizations and Direct Sums of Unit-Endoregular Modules

Part of: Homological methods Modules, bimodules and ideals Rings and algebras arising under various constructions

Published online by Cambridge University Press:  10 August 2018

Xiaoxiang Zhang  and
Gangyong Lee
Xiaoxiang Zhang
Affiliation:
School of Mathematics, Southeast University Nanjing 211189, People's Republic of China ([email protected])
Gangyong Lee
Affiliation:
Department of Mathematics Education, Chungnam National University Daejeon 34134, Republic of Korea ([email protected])
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Abstract

A module is called unit-endoregular if its endomorphism ring is unit-regular. In this paper, we continue the research in unit-endoregular modules. More characterizations of unit-endoregular modules are obtained. As a special case, we show that for an abelian group G, Endℤ(G) is a unit-regular Baer ring if and only if Endℤ(G) is a two-sided extending regular ring. While the class of unit-endoregular modules is not closed under direct sums, we provide a characterization when there are direct sums of two or more unit-endoregular modules also unit-endoregular under certain conditions. In particular, we investigate unit-endoregular modules which are direct sums of indecomposable modules.

Keywords

unit-regular ringunit-endoregular modulestable range 1clean moduletwo-sided extending regular ring

MSC classification

Primary: 16D70: Structure and classification (except as in ), direct sum decomposition, cancellation
Secondary: 16S50: Endomorphism rings; matrix rings 16E50: von Neumann regular rings and generalizations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 4 , November 2018 , pp. 1103 - 1112
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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