EPI-Retractable Modules and Some Applications

Generalizing concepts “right Bezout” and “principal right ideal” of a ring R to modules, an R-module M is called n-epiretractable (resp. epi-retractable) if every n-generated submodule (resp. submodule) ofMR is a homomorphic image ofM. It is shown that if MR is finitely generated quasi-projective 1-epi-retractable, then EndR(M) is a right Bezout (resp. principal right ideal) ring if and only if MR is n-epi-retractable for all n  1 (resp. epiretractable). For a ring R and an infinite ordinal   |R|, the Rmodule M = F  N is epi-retractable where F is a free R-module with a basis set of cardinality  and N is a -generated R-module with . A ring R is quasi Frobenius if every injective R-moduleis epi-retractable. Injective modules in R] are epi-retractable for every N 2 R] if and only if every non-zero factor ring of S is a quasi Frobenius ring where S is an endomorphism ring of a progenerator in R]