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.
The main results of this paper are the following theorems : a) Suppose that C is a chain complex of A− modules with E a non zero subcomplex of C. If E and C/E are hopfians then C is hopfian b) If C is a chain complex of A− modules in which all non trivial subcomplex is cohopfian then C is cohopfian c) If C is a projective chain complex of A− modules with a subcomplex E completely invariant and superfluous then E is hopfian if and only if C/E is hopfian d) If C is an injective chain complex with a subcomplex completely 1904 El Hadj Ousseynou Diallo et al.invariant and essential then E is cohopfian if and only if C is cohopfian e) Any projective and cohopfian or injective and hopfian chain complex of A− modules is a Fitting chain complex.
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