On FGDF-modules

Let R be a unital ring and M a unitary module not necessary over R. The FGDF-module is a generalization of FGDFrings (Touré, Diop, Mohamed and Sangharé, 2014). In this work, we first give some properties of FGDF-modules. After that, we show that for a finitely generated module M, M is a FGDF-module if and only if M is of finite representation type module. Finally, we show that M is a finitely generated FGDF-module if and only if every Dedekind finite module of σ[M] is noetherian.


Introduction
We assume that R is a unity ring and M a unitary module not necessary over R. Let M and N be R-modules.N is said to be generated by M if there exist a set Λ and an epimorphism φ : M (Λ) −→ N. A submodule K of N is said subgenerated by M. The set of submodules of N constitutes the category σ [M].It is a full subcategory of R-Mod whose objects are submodules of a module generated by M (Wisbauer, 1991).
A module M is said to a prime module if for any submodule N of M Ann(N) = Ann(M).A module M is faithful if Ann(M) = 0.A module M is said semisimple if it is direct sum of simple modules.A module M is Hopfian if every surjective endomorphism of M is an automorphism.M is a Dedekind finite module if it is not isomorphic to any proper direct summand of itself.A module M is said to be of finite representation type if it is of finite length and there are only a finite number of non isomorphic finitely generated indecomposable modules in σ[M].A ring R is said to be duo-ring if any one sided ideal is two sided.
The aim of this paper comes from to the following assertion.It is well know that in a commutative ring every finitely generated module is Dedekind finite but the converse is not always true.For instance the Z-module Q is Dedekind finite but not finitely generated.In this paper, we study the modules M for which every Dedekind finite module in σ[M] is finitely generated.Those modules are called FGDF-modules.

Some Properties of FGDF -modules
Lemme 1: (Ghorbani and Haghany, 2002) corollary 1.4 Let R be a ring and M a R-module.If M is Hopfian then it is Dedekind finite.

Proposition 1:
Let R be a ring and M a R-module.If M is a FGDF-module then, there exists a finite number of non-isomorphic simple modules in σ[M].
Proof: Let {N i } I be a complete system of non-isomorphic class of simple modules.Let f : N i −→ N i an epimorphism with f 0 and i ∈ I.
Since each N i is Hopfian and fully invariant then, N is Hopfian.Therefore N is Dedekind finite by lemma 1.Since M is a FGDF-module then, N is finitely generated.Hence I is finite.

Proposition 2:
Let R be a duo ring and M a finitely generated and prime module over End(M).If M is a FGDF-module then, M is a simple.

Proof:
As M is finitely generated, we have an epimorphism f : R → M. It is obvious to see that R/Ann(M) ≃ M by the first theorem of isomorphism.It follows from 15.4 of (Wisbauer, 1991) (Touré, Diop and Sangharé, 2014) that R/Ann(M) is artinian.Hence M is artinian too.Therefore, there exists a simple submodule in M. Let g : R → N be an epimorphism with N the simple submodule of M. Therefore R/Ann(N) ≃ N. Since M is a prime module, then R/Ann(M) = R/Ann(N) is simple.Thus M is simple.

Corollary 1:
Let R be a duo ring and M a finitely generated, prime and faithful module over End(M).If M is a FGDF-module then, R is a field.

Proof:
We have already shown in proposition 2 that R/Ann(M) is isomorphic to a simple module N. As M is a faithful then, Ann(M) = Ann(N) = 0. Hence R is a field.

Proposition 3:
Let M be a module over End(M), then the following conditions are verifyed: (1) The homomorphism image of any FGDF-module is a FGDF-module; (2) Let M = ∏ i∈I M i be a product of its submodules.If M is a FGDF-module then M i is a FGDF-module for each i ∈ I.

The converse is true if for any module N of σ[M] its submodules are fully invariant and σ[M i ]
∩ σ[M j ] = 0 with i j in I finite. Proof: and M is a FGDF-module then K is finitely generated.Hence L is a FGDF-module.
(2) Assume M = ∏ i∈I M i a FGDF-module and f : ∩ σ[M j ] = 0 with i j in I, it follows from proposition 2.2 of (Vanaja, 1996) that As, for each i ∈ I, N i is Dedekind finite then, N i is finitely generated.Hence N = ⊕ i∈I N i is finitely generated since I is finite.It is well know that a homomorphism image, a submodule or a factor of a Dedekind finite module is not in general a Dedekind finite module (Breaz, Cälugäreau and Schulz, 2011)  Proof: It is well-know that the homomorphism image of a finitely generated module is finitely generated then, K is finitely generated.Hence, K is Dedekind finite module.

Proposition 5:
If M is a FGDF-module then, every factor of a Dedekind finite module in σ[M] is a Dedekind finite module.Moreover if M is finitely generated then, every submodule of a Dedekind finite module in σ[M] is Dedekind finite.
Proof: Let N be a Dedekind finite object of σ[M], hence N is finitely generated.Thus for every submodule L of N, N/L is finitely generated, hence a Dedekind finite module.Now let's show that any submodule K of N is a Dedekind finite.Since N is finitely generated and is a module of over R/Ann(M) which is artinian(proposition 2), therefore N is noetherian by 15.21 of (Anderson and Fuller,1974).It is well know that any submodule of noetherian module is finitely generated.Therefore K is finitely generated, hence of Dedekind finite.
but: Proposition 4: If M is a FGDF-module, then the homomorphism image of every Dedekind finite module of σ[M] is a Dedekind finite module.