A module M is called epi-retractable if every submodule of M is a homomorphic image of M. Dually, a module M is called co-epi-retractable if it contains a copy of each of its factor modules. In special case, a ring R is called co-pli (respectively, copri) if RR (respectively, RR) is co-epi-retractable. It is proved that if R is a left principal right duo ring, then every left ideal of R is an epi-retractable R-module. A co-pli strongly prime ring R is a simple ring. A left self-injective co-pli ring R is left Noetherian if and only if R is a left perfect ring. It is shown that a cogenerator ring R is a pli ring if and only if it is a co-pri ring. Moreover, if R is a left perfect ring such that every projective R-module is co-epi-retractable, then R is a quasi-Frobenius ring.

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) of MR is a homomorphic image of M. 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 n³1 (resp. epiretractable). For a ring R and an infinite ordinal b³|R|, the R-module M = FÅN is epi-retractable where F is a free R-module with a basis set of cardinality b and N is a g-generated R-module with g£b. A ring R is quasi Frobenius if every injective R-module is epi-retractable. Injective modules in s[NR] are epi-retractable for every NÎs [MR] 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 s[MR].

We call a module MR essentially retractable if HomR (M,N)¹0 for all essential submodules N of M. For a right FBN ring R, it is shown that: (i) A nonzero module R M is retractable (in the sense that HomR (M,N) ¹0 for all nonzero N£MR) if and only if certain factor modules of M are essentially retractable nonsingular modules over R modulo their annihilators. (ii) A non-zero module MR is essentially retractable if and only if there exists a prime ideal PÎAss (MR) such that HomR (M,N)¹0. Over semiprime right nonsingular rings, a nonsingular essentially retractable module is precisely a module with nonzero dual. Moreover, over certain rings R, including right FBN rings, it is shown that a nonsingular module M with enough uniforms is essentially retractable if and only if there exist uniform retractable R-modules {U i}iÎI and R-homomorphisms M¾®+iÎI U i¾®M with ab;¹0.

Several characterizations of a ring R is given for which any non-zero finitely generated module M is retractable in the sense that HomR(M,N) is non-zero whenever N is a non-zero submodule of M. Such rings are called finite retractable. It is shown that any ring being Morita equivalent to a commutative ring is finite retractable. Also, if the commutative ring is semi-Artinian then any non-zero module is retractable. The class of finite retractable rings is shown to be closed under Morita equivalence and finite direct products. Moreover, for a finite retractable ring R which is a right order in a ring Q, it is shown that Q is also finite retractable.

A quasi-injective weakly co-Hopfian right module over a right Noetherian ring is characterized. Those right semi-Artinian rings and right FBN rings over which completely weakly co-Hopfian right modules are precisely right modules of finite uniform dimension are characterized. In addition, some applications are provided.

We show that every semi-artinian module which is contained in a direct sum of finitely presented modules in s [M], is weakly co-semisimple if and only if it is regular in s [M]. As a consequence, we observe that every semi-artinian ring is regular in the sense of von Neumann if and only if its simple modules are FP-injective.

We investigate the relative cohomology and relative homol-ogy theories of F -Gorenstein modules, consider the relations between classical and F -Gorenstein (co) homology theories.