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Minimally Generated Modules

Published online by Cambridge University Press:  20 November 2018

W. H. Rant
W. H. Rant*
Affiliation:
Department of Natural Sciences Mathematics Lincoln University, 1978, Jefferson City, Missouri 65101
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Abstract

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A non-zero module M having a minimal generator set contains a maximal submodule. If M is Artinian and all submodules of M have minimal generator sets then M is Noetherian; it follows that every left Artinian module of a left perfect ring is Noetherian. Every right Noetherian module of a left perfect ring is Artinian. It follows that a module over a left and right perfect ring (in particular, commutative) is Artinian if and only if it is Noetherian. We prove that a local ring is left perfect if and only if each left module has a minimal generator set.

Keywords

161016401650Minimal generator setperfect ringNoetherian moduleArtinian modulesocleinjective modulefinitely generated
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 23 , Issue 1 , 01 March 1980 , pp. 103 - 105
Copyright
Copyright © Canadian Mathematical Society 1980

References

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