Hopfian and Strongly Cohopfian Objects in the Category of Complexes of Left A-Modules

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. c)Given a chain complex C, if C is quasi-projective and strongly-cohopfian then C is strongly hopfian. In conclusion the main result of this article is the following theorem: Any quasi-projective and strongly-hopfian or quasi-injective and strongly-cohofian chain complex of A-modules is a Fitting chain complex.


Introduction
In this paper, we will study strongly hopfian, strongly cohopfian, quasi-injective, quasi-projective objects of the category of complexes of A-modules denoted by COMP.
We also study Fitting objects of the category of complexes of A-modules.
The objects of COMP are chain complexes and the morphisms are maps of chains.
A chain map f of (C, d) into (C , d ) is defined by: (C, d) : . . .
Given (C, d) an object of COMP and f an endomorphism of (C, d).
C is said to be hopfian (respectively cohopfian) if any epimorphism (respectively monomorphism) is an isomorphism.
→ . . . is said to be fully invariant if and only if each A− module C n is fully invariant, ∀n ∈ Z.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 is direct summand then E and C/E are both strongly Hopfian (respectively strongly coHopfian) then C is strongly Hopfian (respectively strongly coHopfian).b) If E is a fully invariant chain complex such that E and C/E are strongly Hopfian (respectively strongly coHopfian) then C is strongly Hopfian (respectively strongly coHopfian).c) Given C = ⊕C i such C i is fully invariant, if C i is a strongly hopfian (respectively strongly cohopfian) chain complex then C is a strongly hopfian (respectively strongly cohopfian) chain complex.d) Given a chain complex C: . . .
f) Given a chain complex C, if C is quasi-injective and strongly-hopfian then C is strongly cohopfian.g) Given a chain complex C, if C is quasi-projective and strongly-cohopfian then C is strongly hopfian.h) Any quasi-projective and strongly-hopfian or quasi-injective and strongly-cohofian chain complex of A-modules is a Fitting chain complex.
In this paper A denotes a not inevitably commutative, unitarian associative ring and M a left unifere module.

Definitions and Preliminary Results on the Category COMP
Definition 1 Given (C, d) a chain complex of left A modules and f an endomorphism of (C, d) such that: We also define ( Proposition 1 Considering (C, d) a chain complex of A− modules and f an endomorphism of (C, d): where and is the induced by Δ k+1 n+1 .Proposition 2 Let be (C, d) a chain complex of A-modules and f an endomorphism of (C, d) such that: Then δ k+1 n+1 is the induced morphism by

Strongly Hopfian and Strongly CoHopfian
Definition 5 Let be f an endomorphism of chain complex (C, d): Proposition 3 Given f an endomorphism of a chain complex (C, d): So (Ker f k ) stabilizes for all positive integer k.
Proposition 4 Given f an endomorphism of a chain complex (C, d): Definition 6 An A− module M is said to be strongly hopfian (respectively strongly cohopfian) if for any endomorphism f of M the sequence (Ker Definition 7 A chain complex C of A-modules is said to be strongly hopfian(respectively strongly cohopfian), if for any endomorphism f of (C, d), (Ker f k ) (respectively (Im f k )) stabilizes.
Using the Proposition 2.5 (see Hmaimou, 2007) and let be k ∈ N such that Ker f k n ∩ Im f k n = 0 then (Ker f k n ) stabilizes for all n ∈ Z.That prove (Ker f k ) stabilizes, so C is a strongly hopfian chain complex.
Proposition 6 A chain complex C: is strongly cohopfian, if it exists a positive integer k such that: Proof.Using the Proposition 2.6 (see Hmaimou, 2007), we can suppose it exists a positive integer k such that: Theorem 1 Considering C a chain complex and E a subcomplex of C. If C is strongly hopfian (respectively is strongly cohopfian) and E is direct summand then E et C/E both are strongly hopfian (respectively strongly cohopfian).
Proof.Let us demonstrate at first that E is strongly hopfian (respectively strongly cohopfian).
Suppose C = E ⊕ K and let be g a chain map of E in itself which can be extended to C such as f = g ⊕ 0, with 0 the zero morphism of K.
Given C is strongly hopfian (respectively strongly cohopfian), then (Ker , therefore C n is strongly hopfian (respectively strongly cohopfian) and E n is direct summand.
Hence C n and C n /E n for all n ∈ Z are both strongly hopfian (respectively strongly cohopfian).
That prove E and C/E are strongly hopfian (respectively strongly cohopfian).
Theorem 2 Considering C a chain complex and E a subcomplexe of C, if E is fully invariant with E and C/E both strongly hopfian (respectively strongly cohopfian) then C is strongly hopfian (respectively strongly cohopfian).
Proof.Let be (C, d) a chain complex of A-modules and f an endomorphism of (C, d) verifying: f induces also a chain map g of C/E in itself such that: m , so C is strongly hopfian.Corollary 1 Let be C = ⊕C i where C i is a subcomplex fully invariant of C for all i ∈ I.If C i is strongly hopfian (respectively strongly cohopfian), then C is strongly hopfian (respectively strongly cohopfian).
Proof.Let be f an endomorphism of chain complex C. Then it exists a unique family ( f i ) such that i ∈ I where

Quasi-Injective Chain Complex
Definition 8 An A-module M is quasi-injective, if for any monomorphism g of A-module N into M and for any morphism f of N into M, there exists an endomorphism h of M such: Let us considering two chains complexes of A-modules: C and E. C is said to be quasi-injective, if for any monomorphism g of E into C and for any chain map f of E into C, there exists a chain map

. a chain complex of A-modules. C is quasi-injective, if and only if for all n ∈ Z, C n a quasi-injective A-module.
Proof.Suppose that R is quasi-injective chain complex and f : M → N a monomorphism of A− modules and φ n : M → R n a morphism of A− modules.
Considering S and R two chains complexes of A− modules andα a chain of S into R such: where for all n ∈ Z. S n = M and u n = Id M with R n = N and v n = Id N α n is a monomorphism of A− modules, for all n ∈ Z.
Let φ a chain map of S into R such: where for any morphismes Given that R is quasi-injective then it exists ψ a chain map: Reciprocally suppose that for all n ∈ Z, R n is an A-quasi-injective module and let us prove that R quasi-injective chain complex.
Let be γ a monomorphism of chains complexes: Considering β a chain map of complexes such: Let be λ such: R : . . .
Let us demonstrate λ is a chain map: We have: , for all n ∈ Z. so λ is achain map.
Let us verify that: λ • γ = β.We know that for all n ∈ Z, Theorem 4 Given C a chain complex of A-modules.If C is quasi-in jective and strongly hop f ian, then C is strongly cohop f ian.
Proof.Suppose that C is quasi-injective and strongly hopfian, then C n is quasi-injective using Theorem 3 and (Im f k n ) stabilizing this implies that C n is quasi-injective and (ker f k ) stabilizes, so C is strongly cohopfian.Vol. 6, No. 3;2014

Quasi Projective Chain Complex
Definition 10 An A-module P is said to be quasi-projective if for any A-module N and any epimorphism π: P → N and any homomorphism φ: P → N, here exists an endomorphism ψ: Given E is quasi-projective then it exists a chain map ψ such: Reciprocally suppose that for all n ∈ Z, E n is quasi-projective A− and let us demonstrate that E is quasi-projective chain complex.
Considering γ an epimorphism such: Considering λ such: Let us verify that: γ • λ = β.We know for all n ∈ Z, What justifies E is a quasi-projective chain complex.
Theorem 6 Given C a chain complex of A-modules.If C is quasi-pro jective and strongly cohop f ian then C is strongly hop f ian.
Proof.Suppose that C is quasi-projective and strongly cohop f ian, then C n is quasi-projective using Theorem 4 and (ker f k n ) stabilizing this implies that C n is quasi-projective and (ker f k ) stabilizes so C is strongly hopfian.Theorem 8 Any quasi-pro jective and strongly-hop f ian or quasi-in jective and strongly-coho f ian chain complex of A-modules is a Fitting chain complex.

Fitting Chains Complexes
Proof.Suppose that C is quasi-pro jective and cohop f ian then using the previous theorem we can say C is hop f ian, so C is cohop f ian and hop f ian, then is a FIT T ING chain complex.

A→
chain complex (C, d): . . . . . . is said essential if and only if each A− module C n is essential, ∀n ∈ Z.A chain complex (C, d) is said Fitting if for any endomorphism f of (C, d), there exists an integer n such C = Im f n ⊕ Ker f n .After (C, d) is noted by C.

Definition 12 →
An A-module M is said to be FIT T ING module if for any endomorphism f of M, there exists a positive integer n ≥ 1 such:M = Ker f n ⊕ Im f n .Definition 13 A chain complex of A-modules is said to be FIT T ING chain complex if for any endomorphism f of C, it exists n ≥ 1 such C = Ker f n ⊕ Im f n .Theorem 7 Considering C a chain complex of A-modules such C: . . . . ... C is a FIT T ING chain complex if and only if for all n ∈ Z, C n is A-FIT T ING module.Proof.Suppose that C is a FIT T ING chain complex of A-modules then it exists an positive integer k such C = Ker f k ⊕ Im f k donc Ker f k Im f k = 0 and C = Ker f k + Im f k hence for all n ∈ Z, C n = Ker f k n + Im f k n so C n isstrongly hopfian and strongly cohpfian then C n is a FIT T ING A-module.Reciprocally suppose that C n is an A-FIT T ING module then (Ker f k n ) and (Im f k n ) stabilizes so (Ker f k ) stabilizes and (Im f k ) also.Which prove that C is a FIT T ING chain complex.
Given E and C/E are both strongly cohopfian, hence Imh n = Imh n+k and Img n = Img n+k , therefore for all m ∈ Z, we have Imh n m = Imh n+k m and Img n m = Img n+k m .If p = 2n, for x ∈ C m , we obtain g n m 1 / / . . . .
so C is strongly cohopfian.Suppose that E and C/E are both hopfian, then Kerh n = Kerh n+k and Kerg n = Kerg n+k , hence for all m ∈ Z, we Definition 11 Given C and E two chains complexes of A-modules.C is said to be quasi-projective chain complex if for any epimorphism: C → E and for any morphism f : C → E, there exisrsa chain map h: C → C verifying: f = g • h, illustrated by the following commutative diagramm: → . ..a chain complex of A-modules.C is quasi-projective if and only if for all n ∈ Z, C n is a quasi-projective module.Proof.Suppose that E is quasi-projective.Considering f : N → M an epimorphism of A-modules and φ n : E n → M a morphisme of A− modules.Let S and E two chains complexes and α a chain map of E into S such: = M et v n = Id M .S n = N and u n = Id N .Given α n an epimorphism of A− modules.Let φ a chain of E into S verifying: Then given for all n ∈ Z, E n is projective and γ n is an epimorphism of A-modules so it exists λ n −1 . . .S : . . ./ / S n+1 u n+1 / / S n u n / / S n−1 / / . . .