On modules in which every finitely generated submodule is a kernel of an endomorphism

We study an R-module M in which every finitely generated submodule of M is a kernel of an endomorphism of M. Such modules are called Co-epi-finite-retractable (CEFR). We also consider CEFR condition on the co-local modules, submodules and factors of a CEFR module and direct sum of CEFR modules. Among other results, we prove that the injective hull of a simple module over a commutative Noetherian ring is uniserial if and only if it is CEFR.We investigate modules over a principal ideal ring, and show that all finitely generated torsion modules over a principal ideal domain are CEFR. Also, we show that every module over a commutative Kothe ring is CEFR. We also observe that a ring R is left pseudo morphic if and only if it is CEFR as a left R-module and we obtain some new properties of left pseudo morphic rings.