Modules in Which Inverse Images of Some Submodules are Direct Summands

Abstract

Let R be an arbitrary ring with identity and M a right R-module with S = End_R(M). Let F be a fully invariant submodule of M. We call M an F-inverse split module if f⁻¹(F) is a direct summand of M for every f ∈ S. This work is devoted to investigation of various properties and characterizations of an F-inverse split module M and to show, among others, the following results: (1) the module M is F-inverse split if and only if M = F ⊕ K where K is a Rickart module; (2) for every free R-module M, there exists a fully invariant submodule F of M such that M is F-inverse split if and only if for every projective R-module M, there exists a fully invariant submodule F of M such that M is F-inverse split; and (3) Every R-module M is Z₂(M)-inverse split and Z₂(M) is projective if and only if R is semisimple.

Key Words:

• F-inverse split module
• Rickart module
• T-Rickart module

2010 Mathematics Subject Classification:

• 16D10
• 16D40
• 16D80

ACKNOWLEDGMENTS

The authors would like to thank the referee for his/her several helpful suggestions to improve the presentation of this article.