New generalizations of lifting modules

In this paper, we call a module M almost I-lifting if, for any element φ∈S=EndR(M), there exists a decomposition rMℓS(φ) =A⊕B such that A⊆φM and φM∩B≪M. This definition generalizes the lifting modules and left generalized semiregular rings. Some properties of these modules are investigated. We show that if f1+fn= 1in S, where fi,s are orthogonal central idempotents, then M is an almost I-lifting module if and only if each fi M is almost I-lifting. In addition, we call a module M π-I-lifting if, for any φ∈S, there exists a decomposition φnM=eM⊕N for some positive integer n such that e2=e∈S and N≪M. We characterize semi-π-regular rings in terms of π-I-lifting modules.Moreover, we show that if M1 and M2 are abelian π-I-lifting modules with Hom R(Mi, Mj) = 0 for i6=j, then M=M1⊕M2is aπ-I-lifting module.