On modules and rings in which complements are isomorphic to direct summands

Abstract

A right R-module M is virtually extending (or CIS) if every complement submodule of M is isomorphic to a direct summand of M, and M is called a virtually C2-module if every complement submodule of M which is isomorphic to a direct summand of M is itself a direct summand. The class of virtually extending modules (respectively, virtually C2-modules) is a strict and simultaneous generalization of extending modules (respectively, unifies extending modules and C2-modules): M is a semisimple module if and only if M is virtually semisimple and C2, and M is an extending module if and only if M is virtually extending and virtually C2. Furthermore, every virtually simple right R-module is injective if and only if R is a right V-ring and the class of virtually simple right R-modules coincides with the class of simple right R-modules. Among other results, we show that (1) if all cyclic sub-factors of a cyclic weakly co-Hopfian right R-module M are virtually extending, then M is a finite direct sum of uniform submodules; (2) every distributive virtually extending module over any Noetherian ring is a direct sum of uniform submodules; (3) over a right Noetherian ring, every virtually extending module satisfies the Schröder-Bernstein property; (4) being virtually extending (VC2) is a Morita invariant property; (4) if $M\oplus E(M)$ is a VC2-module where $E(-)$ denotes the injective hull, then M is injective.

Communicated by Scott Chapman

Keywords:

• Co-Hopfian module
• Osofsky-Smith Theorem
• Schröder-Bernstein property
• square-free module
• virtually extending module
• virtually C2 module
• virtually semisimple module

2020 Mathematics Subject Classification:

• 16L60
• 16D50
• 16E50

Acknowledgments

The authors would like to express its gratitude to the anonymous reviewer for his/her comments and suggestions which improved the quality of this article (in particular, for Corollary 2.14, Lemma 3.7 and Proposition 3.8). Special thanks to Prof. Scott Chapman. This work is dedicated to the memory of Prof. V. A. Artamonov.