A new generalization of t-lifting modules

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 everyt-coclosed submodule N of M, there exists a direct summand K of Msuch that M=N+K and N\cap K\ll K.We prove that a module with (D_۳) property is tCCif 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}^۲(M) and a submodule L of M that both of them are tCC.

Keywords:

T-small submodule , t-coclosed submodule , t-lifting module , tCC- module