EKFN-Modules

: In this paper, we utilize the σ category to introduce EKFN -modules, which extend the concept of the EKFN -ring. After presenting some properties, we demonstrate, under certain hypotheses, that if M is an EKFN -module, then the following equivalences hold: the class of uniserial modules coincides with the class of cu -uniserial modules; EKFN -modules correspond to the class of locally noetherian modules; and the class of CD -modules is a subset of the EKFN -modules.


Introduction
Throughout this paper, R denotes a commutative, associative ring with an identity 1 0, satisfying the ascending chain condition (ACC) on annihilators.Furthermore, all modules considered are unitary left R-modules.The category of all left R-modules is denoted by R-MOD, with σ representing the full subcategory of R-MOD, whose objects are isomorphic to a submodule of an M-generated module.
A module M is termed noetherian (or artinian) if every ascending (or descending) chain of its submodules becomes stationary.A module M is classified as endo-noetherian (or endo-artinian) if, for any family ( f i ) i≥1 of endomorphisms of M, the sequence {Ker( It is immediately evident that every noetherian ring is endonoetherian, although the converse is not universally true.For instance, Q Z is an endo-noetherian module but not noetherian, as evidenced by the increasing sequence: In [1], C. T. Gueye et al. introduced the class of EKFN-rings.These rings satisfy the following condition: every endo-noetherian module is noetherian.The aim of this paper is to extend the notion of EKFN-rings to the σ category.This new class of modules within this category is termed EKFNmodules, signifying that every endo-noetherian object in σ is noetherian. Recall that an R-module M is defined as uniserial if its submodules are linearly ordered by inclusion.Moreover, M is described as serial if it is a direct sum of uniserial modules.The R-module M is classified as cyclic-uniform uniserial (cu-uniserial) if, for every non-zero finitely generated submodule K ⊆ M, the quotient K/Rad(K) is both cyclic and uniform; here, Rad(K) denotes the intersection of all maximal submodules of M. A module M is termed cyclic-uniform serial (cu-serial) if it is a direct sum of cyclic-uniform uniserial modules.An R-module M is virtually uniserial if, for every non-zero finitely generated submodule K ⊆ M, the quotient K/Rad(K) is virtually simple.Similarly, an R-module M is called virtually serial if it is a direct sum of virtually uniserial modules.M is considered locally noetherian if every finitely generated submodule of M is noetherian.A submodule N of M is defined as small in M if, for every proper submodule L of M, the sum N + L M. A module M is described as hollow if every proper submodule of M is small in M.
A module P is projective if and only if for every surjective module homomorphism f : N → M → 0 and every homomorphism g : P → M, there exists a module homomorphism h : a small cover of M and N is flat.A module A is a projective cover of B provided that A is projective and there exists a minimal epimorphism A → B. It is noted that the projective cover of a module does not always exist but is unique when it does.A ring is termed perfect if every module over this ring possesses a projective cover.The singular submodule Z(M) of a module M is the set of elements m ∈ M such that mI = 0 for some essential right ideal I of R.
We define Z M (N), as a dual of the singular submodule, by Z M (N) = {Ker( f ); f : N → U, U ∈ Γ}, where Γ denotes the class of all M-small modules.A module M is termed discrete if it satisfies conditions (D 1 ) and (D 2 ): (D 1 ): For every submodule N of M, there exists a decomposition The structure of our paper is outlined as follows: Initially, we present preliminary results on EKFN-rings and some fundamental properties of the σ category, particularly when M is finitely generated.Subsequently, we characterize the class of EKFN-modules in rings that satisfy the ACC on annihilators.

Preliminary results
Lemma 1. (Theorem 3.10 of [1] ) Let R be a commutative ring.These conditions are equivalent: 1. R is an EKFN-ring.2. R is a artinian principal ideal ring.Lemma 2. ( From 15.4 of [2]) Let R be a ring and M a R-module.These conditions are verified: 1.If M is finitely generated as a module over S = End(M), then σ = R/Ann(M)-Mod.2. If R is commutative, then for every finitely generated R-module M, we have The second point is a consequence of (1) since we have R/Ann(M) ⊂ S = End(M) canonically.□ Lemma 3. (From 15.2 of [2]) For two R-modules M, N the following are equivalent: 1. N is a subgenerator in σ; Utilitas Mathematica Volume 118, 27-32 Recall a module M is a S -module if every hopfian object of σ is noetherian.
Proposition 1.Let M be a R-module.If M is a S -module then M is an EKFN-module.
Proof.Let N be an endo-noetherian module in σ.N is also hopfian because every endo-noetherian module is strongly hopfian and every strongly hopfian module is hopfian.As M is a S -module then N is noetherian.□

Aims results
Proposition 2. Let M be an EKFN-module.Then the homomorphic image of every endo-noetherian module of σ is endo-noetherian.
Proof.Let N be an endo-noetherian module in σ; as M is an EKFN-module, then N is noetherian.Assume that f : N −→ f (N) = K is an homomorphism image of N. It's well-known that homomorphism image of noetherian module is noetherian.Thus K is noetherian and therefore K is endo-notherian.□ Remark 1.In general, neither a submodule nor a quotient of endo-noetherian module is endonoetherian.For example: • Let R be the free ring over Z generated by {x n , n ∈ N}.Then, R is left endo-noetherian but the left ideal I generated by {x n , n ∈ N} infinite direct sum of left ideals I n generate by {x n }, therefore the R-module I is not endo-noetherian.
) is not endo-noetherian.Proposition 3. Let M be an EKFN-module.Then these conditions are verified.
1. Every submodule of an endo-noetherian module of σ is endo-noetherian.2. Every quotient of an endo-noetherian module of σ is endo-noetherian.
Proof.Let M be an EKFN-module.
1. Let N be an endo-noetherian module in σ and P a submodule of N. As M is an EKFN-module, then N is noetherian and so P because every submodule of a noetherian module is noetherian.Therefore P is endo-noetherian.2. If N is an endo-noetherian module of σ then N is noetherian and it is known that every quotient of a noetherian module is neotherian.Therefore every quotient of N is endo-noetherian.

□
Proposition 4. Let R be a commutative ring.We suppose M is finitely generated over S = End(M).If M is an EKFN-module then these conditions are verified: 1. S = End(M) has stable range 1 and codim(End(M)) ≤ dim(M) + codim(M).2. There are, up to isomorphim, only many finitely indecomposable projective End(M)-modules. Proof.
. As End(M) is semilocal then by theorem 4.10 of[3]we have the result.□semisimpletheneveryobject of σ[M/Ann(M)] is semisimple therefore N is semisimple.It is well known that for a semisimple module, endo-noetherian and noetherian coincide.In conclusion M is an EKFN-module.□RecallM is locally noetherian if every finitely generated submodule of M is noetherian.Lemma 4. (From 27.3 of[2]) For an R-module M the following assertions are equivalent:Hence R/Ann(M) is noetherian; M ≃ R/Ann(M) is finitely generated and noetherian.Morever N ∈ σ implies N is an ideal of R/Ann(M); hence a submodule of M. It's well know over noetherian ring, every submodule of finitely generated module is finitely generated.So N is noetherian because over noetherian ring, finitely generated and noetherian coincide.Therefore M is an EKFN-module.□ 1. M finitely generated EKFN-module over S = End(M) implies M is artinian, hence M has a finite Goldie dimension.In addition M is cohopfian means every injective endomorphism of M is bijective.Referring to Theorem 4.3 of [3], then the endomorphism ring S = End(M) is semilocal.Therefore S = End(M) has stable range 1. 2