SS-LIFTING MODULES AND RINGS.

A moduleM is called ss-lifting if for every submodule A of M, there is a decomposition M = M1 ⊕ M2 such that M1 ≤ A and A∩M2 ⊆ Socs (M), where Socs(M) = Soc(M)∩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, RR 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) ⊆ Soc(RR) iff every left R-module is ss-lifting. We also study on factor modules of ss-lifting modules. [ABSTRACT FROM AUTHOR]