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.
The main results of this paper are: 1. Lets A = n∈Z A n be a graded domain and S is the multiplicative set satisfying the left conditions of Ore formed of homogeneous elements of A, then we have: The relation S −1 Gr (−) : AG r − M od −→ S −1 Gr A − M od which that for any graded left A−module M we correspond the graded left S −1 Gr (A)-module S −1 Gr (M) and for all graded morphism of graded left A−modules f : M −→ N we correspond the graded morphism S −1 Gr (f) of graded left S −1 Gr (A)-modules is a exact covariant functor that preserves degrees; 2. Lets A = n∈N A n be a positively graded domain and S is the multiplicative set satisfying the left conditions of Ore formed of homogeneous elements of A, M = n∈N M n , N = n∈N N n and L = n∈N L n three positively graded left A−modules then :
The object of this paper is the study of strongly hopfian, strongly cohopfian, quasi-injective, quasi-projective, Fitting objects of the category of complexes of A-modules. In this paper we demonstrate the following results: a)If C is a strongly hopfian chain complex (respectively strongly cohopfian chain complex) and E a subcomplex which is direct summand then E and C/E are both strongly Hopfian (respectively strongly coHopfian) then C is strongly Hopfian (respectively strongly coHopfian). b)Given a chain complex C, if C is quasi-injective and strongly-hopfian then C is strongly cohopfian.
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