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.