A KIND OF F-INVERSE SPLIT MODULES

Let M be a right module over a ring R. In this manuscript, we shall study on a special case of F-inverse split modules where F is a fully invariant submodule of M introduced in [12]. We say M is Z^2 (M)-inverse split provided f −1 (Z^2 (M)) is a direct summand of M for each endomorphism f of M. We prove that M is Z^2 (M)-inverse split if and only if M is a direct sum of Z^2 (M) and a Z^2 -torsionfree Rickart submodule. It is shown under some assumptions that the class of right perfect rings R for which every right R-module M is Z^2 (M)-inverse split (Z(M)-inverse split) is precisely that of right GV -rings.