X-LIFTING MODULES OVER RIGHT PERFECT RINGS

Abstract

Keskin and Harmanci defined the family B(M, X) = {A ≤ M | ∃Y ≤ X, ∃f ∈ HomR(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 B(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 P contains Rad(P ), 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 ∈ B(H, X), if H ⊕H has the internal exchange property, then H has a local endomorphism ring.