SS-Lifting Modules and Rings

Abstract

A module M is called ss-lifting if for every submodule A of M, there is a decomposition M = M-1 circle plus M-2 such that M-1 <= A and A boolean AND M-2 subset of SOCs (M), where Soc(s) (M) = Soc(M) boolean AND Rad(M). In this paper, we provide the basic properties of ss-lifting modules. It is shown that: (1) a module M is ss-lifting iff it is amply ss-supplemented and its ss-supplement submodules are direct summand; (2) for a ring R, R-R is ss-lifting if and only if it is ss-supplemented iff it is semiperfect and its radical is semisimple; (3) a ring R is a left and right artinian serial ring and Rad (R) subset of Soc (R-R) iff every left R-module is ss-lifting. We also study on factor modules of ss-lifting modules.