We study in this work the notions of hopficity and cohopficity in the categories AGr(A - Mod) and COMP(AGr(A - Mod)) of associate complex to a graded left A-module and we show that: 1. Let M a graded left A-module, N a graded submodule of M, M_* be a complex associate to M. Suppose that M_* be a quasi-projective and N be a completely invariant and essential sub-complex of M_* associate to N. Then N_* is cohopfian if, and only, if M_* is cohopfian. 2. Let M a graded left A-module, N a graded submodule of M, M_* quasi-injective and M_* a completely invariant and superfluous sub-complex of M_*. Then M_* is cohopfian if, and only, if M_*=M_* is cohopfian.
Let M be a graded leftA-module and M∗ the associate complex of M. Then : If M∗ is noetherian (resp. artinian) then M∗ is strongly hopfian (resp. strongly cohopfian); If M∗ is strongly hopfian (resp. cohopfian), then M∗ is hopfian (resp. cohopfian); Let M be a graded leftA-module, M∗ the associate complex of M, N a submodule of M, and N∗ fully invariant subcomplex of M∗. Then : If N∗ and M∗/N∗ strongly hopfian, then M∗ is strongly hopfian. Let M a graded leftA-module, N a submodule of M and M∗ the associate complex of M. Then : if all subcomplex of M∗ is cohopfian, then M∗ is cohopfian. if M∗/N∗ is strongly hopfian, then M∗ is strongly hopfian
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.
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