A new generalization of t-lifting modules

Abstract

In this paper we introduce the concept of $tCC$-modu-les which is a proper generalizationof ($t$-)lifting modules. Let $M$ be a module over a ring $R$.We call $M$ a $tCC$-module(related to $t$-coclosed submodules) provided that for every$t$-coclosed submodule $N$ of $M$, there exists a direct summand $K$ of $M$such that $M=N+K$ and $Ncap Kll K$.We prove that a module with $(D_3)$ property is $tCC$if and only if every direct summand of $M$ is $tCC$. It is also shownthat an amply supplemented module $M$ is $tCC$ if and only if $M$ decomposed to$overline{Z}^2(M)$ and a submodule $L$ of $M$ that both of them are $tCC$.