Abstract

Keskin and Harmanci defined the family <TEX>$\mathcal{B}(M,X)=\{A{\leq}M{\mid}{\exists}Y{\leq}x,{\exists}f{\in}Hom_R(M,X/Y),Ker\;f/A{\ll}M/A\}$</TEX>. And Orhan and Keskin generalized projective modules via the class <TEX>$\mathcal{B}$</TEX>(M,X). In this note we introduce X-local summands and X-hollow modules via the class <TEX>$\mathcal{B}$</TEX>(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 <TEX>$K{\in}\mathcal{B}(H,X)$</TEX>, if <TEX>$H{\oplus}H$</TEX> has the internal exchange property, then H has a local endomorphism ring.