X-LIFTING MODULES OVER RIGHT PERFECT RINGS

Keskin and Harmanci defined the family beta(M, X) = {A <= M vertical bar EY <= X, Ef is an element of Hom(R)(M, X/Y),Ker f/A<< M/A}. And Orhan and Keskin generalized projective modules via the class B(M, X). In this note we introduce X-local summands and X-hollow modules via the class beta(M, X). Let R be a right perfect ring and let M be an X-lifting module. We prove that if every co-closed submodule of any projective module contains its radical, then M has an indecomposable decomposition. This result is a generalization of Kuratomi and Chang's result [9, Theorem 3.4]. Let X be an R-module. We also prove that for an X-hollow module H such that every non-zero direct summand K of H with K is an element of beta(H, X), if H circle plus H has the internal exchange property, then H has a local endomorphism ring.