On modules in which every proper finitely generated submodule is a subset of a kernel of a nonzero endomorphism

We study an R-module M such that for every proper finitely generated submodule K of M, Hom R(M/K,M)≠0. Such modules are called co-finite-retractable (simply CFR). We consider when factor modules, direct sums and direct summands of CFR modules are CFR. Also, we investigate the injective hull of CFR modules. We explore certain properties of a ring R such that every (cyclic, free) module is CFR. Also, we show that R is a semisimple ring if and only if every R-module is regular if and only if every CFR module over R is regular.