Localization of Hopfian and Cohopfian Objects in the Categories of A − Mod, AGr(A − Mod) and COMP(AGr(A − Mod))

The aim of this paper is to study the localization of hopfian and cohopfian objects in the categories A-Mod of left A-modules, AGr(A-Mod) of graded left A-modules and COMP(AGr(A-Mod)) of complex  sequences associated to graded left A-modules.We have among others  the main following results :1. Let M be a noetherian graded left A-module, S a saturated multiplicative part formed by the non-zero homogeneous elements of A verifying the left Ore conditions, N a submodule of M, M_{*} is a noetherian quasi-injective complex sequence associated with M and N_{*} is an essential and completely invariant complex sub\--sequence of M_{*}. Then, S^{-1}(N_{*}) the complex sequence of morphisms of left S^{-1}A\--modules is cohopfian if, and only, if S^{-1}(M_{*}) is cohopfian ;2. let M be a graded left A\--module and S a saturated multiplicative part formed by the non-zero homogeneous elements of A verifying the left Ore conditions. If M_{*} is a hopfian, noetherian and quasi-injective complex sequence associated with M, then the complex sequence of morphisms of left S^{-1}(A)-modules S^{-1}(M_{*}) has the following property :{any epimorphism of sub-complex S^{-1}(N_{*}) of S^{-1}(M_{*}) is an isomorphism } ;3. let M be a graded left A-module, N a graded submodule of M, S a saturated multiplicative part formed by the non-zero homogeneous elements of A verifying the left Ore conditions. M_{*} the quasi-projective complex sequence associated with M and $N_{*}$ a superfluous and completely invariant complex sub\--sequence of $M_{*}$. Then the complex morphism sequence of left $S^{-1}(A)$\--modules $S^{-1}(N_{*})$ is hopfian if, and only if, $S^{-1}(M_{*}/N_{*})$ the complex sequence associated with S^{-1}(M/N) is hopfian.