a generalization of coretractable modules

Let r be a ring and m a right r-module. we call m , coretractable relative to z(m ) (for short, z(m )-coretractable) provided that, for every proper submodule n of m containing m z(m ), there exists a nonzero homomorphism f : n → m . we investigate some conditions under which two concepts of coretractable and z(m )-coretractable, coincide. for a commutative semiperfect ring r, we show that r is z(r)-coretractable if and only if r is a kasch ring. some examples are provided to illustrate different concepts.