University of Guilan Journal of Algebra and Related Topics 2345-3931 8 1 2020 06 01 A new generalization of t-lifting modules 1 13 4160 10.22124/jart.2020.16482.1203 EN Y. Talebi University of Mazandaran, Babolsar, Iran. A. R. Moniri Hamzekolaee Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran M. Hosseinpour Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran S. Asgari Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar Journal Article 2020 05 07 In this paper we introduce the concept of $tCC$-modu\-les which is a proper generalization<br />of ($t$-)lifting modules. Let $M$ be a module over a ring $R$.<br />We call $M$ a $tCC$-module<br />(related to $t$-coclosed submodules) provided that for every<br />$t$-coclosed submodule $N$ of $M$, there exists a direct summand $K$ of $M$<br />such that $M=N+K$ and $N\cap K\ll K$.<br />We prove that a module with $(D_3)$ property is $tCC$<br />if and only if every direct summand of $M$ is $tCC$. It is also shown<br />that an amply supplemented module $M$ is $tCC$ if and only if $M$ decomposed to<br />$\overline{Z}^2(M)$ and a submodule $L$ of $M$ that both of them are $tCC$. https://jart.guilan.ac.ir/article_4160_3931d1f7db9bb0773c27c6fcea42e273.pdf