M. Hosseinpour - Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, P.O. Box ۴۷۴۱۶-۹۵۴۴۷, Babolsar, Iran.
A. R. Moniri Hamzekolaee - Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, P.O. Box ۴۷۴۱۶-۹۵۴۴۷, Babolsar, Iran.

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)(Z2(M)) is a direct summand of M for each endomorphism f of M. We prove that M is Z2(M)-inverse split if and only if M is a direct sum of Z2(M) and a Z2-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 Z2(M)-inverse split (Z(M)-inverse split) is precisely that of right GV-rings.

Rickart module, Z(M)-inverse split module, Z^2(M)-inverse split module