On FGS -Modules

We consider R a non-necessarily commutative ring with unity 1 0 and M a module over R. By using the category σ[M] we introduce the notion of FGS -module. The latter generalizes the notion of FGS -ring. In this paper we fix the ring R and study M for which every hopfian module of σ[M] becomes finitely generated. These kinds of modules are said to be FGS -modules. Some properties of FGS -module, a characterization of semisimple FGS -module and of serial FGS -module over a duo ring have been obtained.


Introduction
We consider R a non-necessarily commutative ring with unity 1 0 and M a module over R. By using the category σ[M] we introduce the notion of FGS -module.The latter generalizes the notion of FGS -ring (Barry, Sangharé, & Touré, 2007).The set of all modules subgenerated by M is said category σ [M].It is the full subcategory of R-Mod.A module N is hopfian if every R-epimorphism of N is an automorphism.We know every module which is finitely generated is hopfian but the opposite is not true.For instance, we assume Z be the ring of integers, then the module Q of rational numbers over Z is hopfian but Q is not finitely generated.
The goal of this work is: we fix the ring R and study M for which every hopfian module of σ[M] becomes finitely generated.These kinds of modules are said to be FGS -modules.We have obtained, as results, some properties of FGS -module, a characterization of semisimple FGS -module (Theorem 1) and of serial FGS -module (Theorem 2) over a duo ring.

Definitions and Some Properties of FGS -Modules
We consider M and N two objects of R-Mod.We say that N is generated by M if there is a surjective homomorphism Ψ and a set Λ such that Ψ : M (Λ) → N. A submodule of N is said to be subgenerated by M. The set of all submodule of N constitutes the category σ[M].It's the full subcategory of R-Mod.A projective, finitely generated and generator object of σ[M] is said to be progenerator.A module which its submodules are linearly ordered by inclusion is uniserial.A module is homo-uniserial if the factor of two finitely generated submodules with their radical are simple and isomorphic.A module which is a direct sum of uniserial (resp.homo-uniserial) modules is serial (resp.homo-serial).In σ[M] if every object is serial, then M is of serial representation type.A module is of finite representation type, if it is of finite length and there exists only a finite number of nonisomorphic indecomposable modules.A module M is a S -module if every hopfian object of σ[M] is noetherian.
Proposition 1 We consider M a module over R, 1) If M is a FGS -module, then every submodule of M is a FGS -module too; 2) If a module is a FGS -module, then every homomorphic image of that module is a FGS -module; 3) Let consider M a direct product of its submodules.
If the product is finite and for every different submodules N, K of M we have Hom(N, K) = 0, then, the converse of 1) is true. Proof.
3) We assume N an hopfian object of σ[M].Since the product is finite there exists an isomorphism between the product and the direct sum.That implies N ∈ σ[ i∈I M i ].As Hom(M i , M j ) = 0 for every i j then by referring to Vanaja (1996), N = i∈I N i and N i ∈ σ[M i ], i ∈ I.Then, for any i ∈ I, N i is finitely generated since N i is hopfian.Therefore N is finitely generated.Thus M is a FGS -module.

Proposition 2 We consider M a module. If M is a FGS -module hence, we will have a finite number of module which are non-isomorphic simple objects in σ[M].
Proof.We assume (N λ ) λ∈Λ be a complete system of non-isomorphic class of simple object of σ[M].We put As N is hopfian then N is finitely generated.Thus Λ is finite.

Proposition 3 We consider M a module. If M is FGS -module then every indecomposable projective object of σ[M] is finitely generated.
Proof.We assume N a projective object of σ[M].We consider also f an endomorphism of N. We assume the following exact sequence: Therefore N is hopfian.That implies N is finitely generated.
Proposition 4 We consider M a module.If M is a FGS -module then, the projective cover of every simple module of σ[M], if it exists, is finitely generated.
Proof.We assume P a simple module of σ[M] with projective cover P. To show that the projective cover is finitely generated it suffices to show that P is indecomposable.
Let P 1 and P 2 be two submodules of P. We suppose P = P 1 ⊕ P 2 and a surjective homomorphism f : P → P such that ker( f ) is superfluous in P. Let f 1 0 be the restriction of f on P 1 .As P is simple then f 1 is surjective.Then P 2 ⊆ ker( f ).That implies P 2 = {0}.Therefore P is indecomposable.It results from Proposition 3, P is finitely generated.
A module which is finitely generated and homo-uniserial module is uniserial.
Proposition 5 We consider R be a duo ring and M a FGS -module.If M is of homo-serial representation type such that σ[M] has a progenerator then, M is of serial representation type.
Proof.We suppose N ∈ σ[M] and Q the progenerator of σ[M].We consider x ∈ Q.We have σ[Rx] = R/I-Mod where I = Ann(x).As the Rx is a FGS -module then the factor ring R/I is a FGS -ring.By referring to (Barry, Sangharé, & Touré, 2007, Theorem 3.4) R/I is an artinian principal ideal ring.As σ[Q] = σ[M] then, from Corollary 3.5 every module of σ[M] is a direct sum of cyclic modules.We assume N be an homo-serial object of σ[M].Hence N = n i=1 N i where N i is homo-uniserial.Therefore, for every 1 ≤ i ≤ n, N i is cyclic.Then N i is finitely generated for every 1 ≤ i ≤ n.N i is uniserial, for every 1 ≤ i ≤ n.Hence N is serial.Thus M is of serial representation type.
Remark 1 The converse of the Proposition 5 has been done in Wisbauer (1991).
Proposition 6 We consider R a duo ring and M a serial module.We assume that σ[M] has a progenerator and Hom(N, K) = 0 for any submodules N et K of M. We have the equivalence between the next assertions: 1) R is a FGS -ring; 2) M is a FGS -Module.

Proof.
1) ⇒ 2) We assume N an hopfian object of σ[M].As N is also an object of R-Mod and R is FGS ring hence, N is