Simple-direct-injective modules

Victor Camillo; Yasser Ibrahim; Mohamed Yousif; Yiqiang Zhou

doi:10.1016/j.jalgebra.2014.07.033

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Simple-direct-injective modules

Victor Camillo, Yasser Ibrahim, Mohamed Yousif and Yiqiang Zhou

Journal of algebra, Vol.420, pp.39-53

12/15/2014

DOI: 10.1016/j.jalgebra.2014.07.033

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Abstract

A module M over a ring is called simple-direct-injective if, whenever A and B are simple submodules of M with A≅B and B⊆⊕M, we have A⊆⊕M. Various basic properties of these modules are proved, and some well-studied rings are characterized using simple-direct-injective modules. For instance, it is proved that a ring R is artinian serial with Jacobson radical square zero if and only if every simple-direct-injective right R-module is a C3-module, and that a regular ring R is a right V-ring (i.e., every simple right R-module is injective) if and only if every cyclic right R-module is simple-direct-injective. The latter is a new answer to Fisher's question of when regular rings are V-rings [8].

Artinian serial ring

C2-module

C3-module

Injective module

Regular ring

Simple-direct-injective module

V-ring

Details

Title: Subtitle: Simple-direct-injective modules
Creators: Victor Camillo - University of Iowa
Yasser Ibrahim - Cairo University
Mohamed Yousif - The Ohio State University
Yiqiang Zhou - Memorial University of Newfoundland
Resource Type: Journal article
Publication Details: Journal of algebra, Vol.420, pp.39-53
DOI: 10.1016/j.jalgebra.2014.07.033
ISSN: 0021-8693
eISSN: 1090-266X
Publisher: Elsevier Inc
Grant note: DOI: 10.13039/100006928, name: Ohio State University; DOI: 10.13039/501100000038, name: NSERC, award: 194196
Language: English
Date published: 12/15/2014
Academic Unit: Mathematics
Record Identifier: 9984241052102771

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