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 $X$-lifting modules over right perfect rings Bull. Korean Math. Soc. 2008 Vol. 45, No. 1, 59-66 Printed March 1, 2008 Chaehoon Chang Kyungpook National University Abstract : 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 Downloads: Full-text PDF

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