Bulletin of the
Korean Mathematical Society BKMS

Keskin and Harmanci defined the family $\mathbf{B}$$(M,X)$ $=\{A\leq M \mid \exists Y \leq X, \exists f \in \mbox{Hom}_{R}(M,X/Y), {\rm Ker}$ $f/A \ll M/A\}$. And Orhan and Keskin generalized projective modules via the class $\mathbf{B}$$(M,X)$. In this note we introduce $X$-local summands and $X$-hollow modules via the class $\mathbf{B}$$(M,X)$. Let $R$ be a right perfect ring and let $M$ be an $X$-lifting module. We prove that if every co-closed submodule of any projective module $P$ contains $\mbox{Rad}(P)$, then $M$ has an indecomposable decomposition. This result is a generalization of Kuratomi and Chang's result [9, Theorem 3.4]. Let $X$ be an $R$-module. We also prove that for an $X$-hollow module $H$ such that every non-zero direct summand $K$ of $H$ with $K \in$ $\mathbf{B}$$(H,X)$, if $H \oplus H$ has the internal exchange property, then $H$ has a local endomorphism ring.

Keywords: right perfect ring, lifting module, exchange property

MSC numbers: Primary 16D40, 16P70