A generalization of ⊕ ⊕ -cofinitely supplemented modules

ýWe say that a module M M is a \emph{cms-module} ifý, ýfor every cofinite submodule N N of M M ý, ýthere exist submodules K K and K ′ K′ of M M such that K K is a supplement of N N ý, ýand K K ý, ýK ′ K′ are mutual supplements in M M ý. ýIn this articleý, ýthe various properties of cms-modules are given as a generalization of ⊕ ⊕ -cofinitely supplemented modulesý. ýIn particularý, ýwe prove that a π π -projective module M M is a cms-module if and only if M M is ⊕ ⊕ -cofinitely supplementedý. ýFinallyý, ýwe show that every free R R -module is a cms-module if and only if R R is semiperfect.