Co-retractable Modules and Semi-Simplicity Property

Abstract

Co-retractable modules appeared in 2006 by Clark, Lomp and Vanaja. In this article, we present and study the relationships between two known concepts namely semi-simple modules and Co-retractable module. Our concern in this article is to develop some properties of Co-retractable module and related to other classes of modules such as multiplication module, indecomposable module and projective module. We prove that if M is a multiplication R-module with annR(M) is a prime ideal and the Socal of M is a direct summand of M, this means M is a retractable module. Also, if M is an indecomposable multiplication R-module with annR(M) is a prime ideal of Noetherian Ring R and every direct sum of modules with summand intersection property has summand sum property, then M is a co-retractable module. Finally, if M is a simple quasi-Dedekind module over the Artinian Ring R with rad(R) = 0 and annR(M) is a prime ideal of R, then M is a co-retractable module.