After the paper [1] was published, following information about D-extending modules pointed out by Adnan Tercan. Let $r$ be a left exact preradical in category of right $R$-modules and let $M$ $R$-module. A submodule $N$ is called provided $r(M/N)=0$. module an ES-module every direct summand $M$. ES-modules introduced A. Tercan [2]. Since socle preradical, coincide with for $r=Soc$.The author wh...

A module M is said to have the SIP if intersection of each pair direct summands also a summand M. In this article, we define SIPr and only exact where r left preradical for category right modules. We investigate structural properties SIPr-modules locate implications between other properties. deal with decomposition theory as well SIPr-modules. provide examples by looking at special preradicals.

For a coalgebraC , the rational functor Rat(−) : C∗ → C∗ is a left exact preradical whose associated linear topology is the family C , consisting of all closed and cofinite right ideals of C∗. It was proved by Radford (1973) that if C is right Noetherian (which means that every I ∈ C is finitely generated), then Rat(−) is a radical. We show that the converse follows if C1, the second term of th...

In this paper the notion of Rees short exact sequence for S-posets is introduced, and we investigate the conditions for which these sequences are left or right split. Unlike the case for S-acts, being right split does not imply left split. Furthermore, we present equivalent conditions of a right S-poset P for the functor Hom(P;-) to be exact.

‎A module $M$ is called FI-extending if every fully invariant submodule of $M$ is essential in a direct summand of $M$‎. ‎It is not known whether a direct summand of an FI-extending module is also FI-extending‎. ‎In this study‎, ‎it is given some answers to the question that under what conditions a direct summand of an FI-extending module is an FI-extending module?