Virtually homo-uniserial modules and rings

Abstract

We study the class of virtually homo-uniserial modules and rings as a nontrivial generalization of homo-uniserial modules and rings. An R-module M is virtually homo-uniserial if, for any finitely generated submodules $0\ne K,L\subseteq M,$ the factor modules $K/\text{Rad}(K),$ and $L/\text{Rad}(L)$ are virtually simple and isomorphic (an R-module M is virtually simple if, $M\ne 0$ and $M\cong N$ for every nonzero submodule N of M). Also, an R-module M is called virtually homo-serial if it is a direct sum of virtually homo-uniserial modules. We obtain that every left R-module is virtually homo-serial if and only if R is an Artinian principal ideal ring. Also, it is shown that over a commutative ring R, every finitely generated R-module is virtually homo-serial if and only if R is a finite direct product of almost maximal uniserial rings and principal ideal domains with zero Jacobson radical. Finally, we obtain some structure theorems for commutative (Noetherian) rings whose every proper ideal is virtually (homo-)serial.

Keywords:

• Homo-uniserial module
• virtually simple module
• virtually homo-uniserial module

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 16D70
• 16D60
• Secondary 13A18
• 13F30

Acknowledgments

The authors would like to thank the anonymous referee for a careful checking of the details and for helpful comments that improved this paper.

Additional information

Funding

The research of the first and the second authors was in part supported by grants from IPM (No. 99130214 and No. 99160418). This research is partially carried out in the IPM-Isfahan Branch.