On 〖Rad〗_R*-lifting module

Abstract

The goal of this study is to introduce a new concept on a lifting module. Let M be a unitary left  R-module, and let R be any ring with one. M is called 〖Rad〗_R*-lifting for short (R^*-lifting) for any A ≤ M, there are some submodule B of A that is M=B⨁B_1, B_1≤M and A∩B_1 ≪_R*B_1.We prove some characterization of this class of modules. We show that every R^*-hollow is R^*-lifting also. The quotient of R^*-lifting module is R^*-lifting under some conditions. Some results about this concept are given. The relation between this concept and other modules related with it are presented. In addition, FI-R^*-lifting was introduced as fallows: if for any submodule fully invariant A≤ M , there is some submodule D of M that is M=D⨁D_1,D_1≤M and A∩D_1 ≪_R*D_1.it was proved that for any direct summand of   R^*-supplement is R^* lifting module.