Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2017; 32(2): 277-285

Online first article September 29, 2016      Printed April 30, 2017

https://doi.org/10.4134/CKMS.c160103

Copyright © The Korean Mathematical Society.

On subdirect product of prime modules

Najmeh Dehghani and Mohammad Reza Vedadi

Persian Gulf University, Isfahan University of Technology

Abstract

In the various module generalizations of the concepts of prime (semiprime) for a ring, the question ``when are semiprime modules subdirect product of primes?" is a serious question in this context and it is considered by earlier authors in the literature. We continue study on the above question by showing that: If $R$ is Morita equivalent to a right pre-duo ring (e.g., if $R$ is commutative) then weakly compressible $R$-modules are precisely subdirect products of prime $R$-modules if and only if ${\rm dim}(R)=0$ and $R/{\rm N}(R)$ is a semi-Artinian ring if and only if every classical semiprime module is semiprime. In this case, the class of weakly compressible $R$-modules is an enveloping for Mod-$R$. Some related conditions are also investigated.

Keywords: classical prime module, prime module, semi-Artinian ring, semiprime module, weakly compressible module

MSC numbers: Primary 16N60, 16D90; Secondary 16P40

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