Prime virtually semisimple modules and rings

Abstract

This article is a sequel to the recent three papers on “virtually semisimple modules and rings,” by Behboodi et al., which two of them appeared in the Algebras and Representation Theory and Communications in Algebra in 2018. An R-module M is called virtually semisimple if each submodule of M is isomorphic to a direct summand. A ring R is called left (resp., right) virtually semisimple if ${}_{R}R$ (resp., R_R) is virtually semisimple. In this article, we study rings and modules in which every prime submodule is isomorphic to a direct summand, and called them prime virtually (or $\wp$-virtually) semisimple modules. A ring R is called left (resp., right) $\wp$-virtually semisimple if ${}_{R}R$ (resp., R_R) is $\wp$-virtually semisimple. The results of the article are inspired by a characterization of left $\wp$-virtually semisimple rings. We prove that these rings are precisely the left virtually semisimple rings, and in this case $R\cong {\prod }_{i=1}^k{M}_{n_i}(D_i)$, where each D_i is a domain and each $M_{n_i}(D_i)$ is a principal left ideal ring. We also answer to the following questions: (i) Describe rings R where each (finitely generated or cyclic) left R-module is $\wp$-virtually semisimple?, and (ii) Describe rings R where each left R-module is a direct sum of indecomposable $\wp$-virtually semisimple modules? Finally, we study $\wp$-virtually semisimple modules over commutative rings.

Keywords:

• Prime submodule
• prime virtually semisimple module
• semisimple module
• semisimple ring
• virtually semisimple module
• virtually semisimple ring
• Wedderburn–Artin theorem

2010 Mathematics Subject Classification:

• Primary 16D60
• 16S50
• 13F10 Secondary 16D70

Acknowledgments

The authors owe a great debt to the referee who has carefully read earlier version of this paper and made significant suggestions for improvement. We would like to express our deep appreciation for the referees work.

Additional information

Funding

The research of the first author was in part supported by a grant from IPM, Isfahan Branch (No. 96130413). This research is partially carried out in the IPM-Isfahan Branch.