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.
Najmeh Dehghani and Mohammad Reza Vedadi
Persian Gulf University, Isfahan University of Technology
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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