EKFN-Modules

DIOMPY, Mankagna; BOUSSO, Ousseynou; DIOUF, Remy Diaga; DIANKHA, Oumar

doi:10.61091/um118-03

Abstract

1. Introduction
2. Preliminary results
3. Aims results
Conflict of Interest

References

Utilitas Mathematica

Volume 118
Pages: 27-32

Research article

EKFN-Modules

^¹, ^¹, ^¹, ^¹

¹Département de Mathématiques et Informatique, Faculté des Sciences et Techniques, Université Cheikh Anta Diop, 5005 Dakar (Senegal).

Received: 22/10/2023
Accepted: 25/12/2023
Published: 08/01/2024

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Abstract

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

Keywords: Noetherian, endo-noetherian, EKFN-module, Uniserial,

c u

-uniserial, Virtually uniserial, Locally noetherian and CD-module

1. Introduction

Throughout this paper, $R$ denotes a commutative, associative ring with an identity $1 \neq 0$ , 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 $(f_{i})_{i \geq 1}$ of endomorphisms of $M$ , the sequence ${K e r (f_{1}) \subseteq K e r (f_{2}) \subseteq \dots}$ (or ${ℑ (f_{1}) \supseteq ℑ (f_{2}) \supseteq \dots}$ , respectively) stabilizes. It is immediately evident that every noetherian ring is endo-noetherian, 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: $Z \subset \frac{1}{2} Z \subset \frac{1}{4} Z \subset \dots$ In [1], C. T. Gueye et al. introduced the class of $E K F N$ -rings. These rings satisfy the following condition: every endo-noetherian module is noetherian. The aim of this paper is to extend the notion of $E K F N$ -rings to the $σ$ category. This new class of modules within this category is termed $E K F N$ -modules, 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 $K \subseteq M$ , the quotient $K / R a d (K)$ is both cyclic and uniform; here, $R a d (K)$ 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 $K \subseteq M$ , the quotient $K / R a d (K)$ 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 $N + L \neq M$ . 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 $f : N \to M \to 0$ and every homomorphism $g : P \to M$ , there exists a module homomorphism $h : P \to N$ such that $f \circ h = g$ . A submodule $N$ of $M$ is termed superfluous if the only submodule $T$ of $M$ satisfying $N + T = M$ is $M$ itself. A morphism $f : A \to B$ of modules is considered minimal provided that $\ker (f)$ is a superfluous submodule of $A$ . A module $N$ is designated a small cover of a module $M$ if there exists a minimal epimorphism $f : N \to M$ . $N$ is called a flat cover of $M$ if $N$ is 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 $A \to B$ . 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 $m \in M$ such that $m I = 0$ for some essential right ideal $I$ of $R$ .

We define ${\bar{Z}}_{M} (N)$ , as a dual of the singular submodule, by ${\bar{Z}}_{M} (N) = ⋂ {K e r (f); f : N \to U, U \in Γ}$ , where $Γ$ denotes the class of all $M$ -small modules. A module $M$ is termed discrete if it satisfies conditions $(D_{1})$ and $(D_{2})$ :

For every submodule $N$ of $M$ , there exists a decomposition $M = M_{1} \oplus M_{2}$ such that $M_{1} \subseteq N$ and $M_{2} \cap N$ is small in $M_{2}$ .
For any summand $K$ of $M$ , every exact sequence $M \to K \to 0$ splits.

The structure of our paper is outlined as follows: Initially, we present preliminary results on $E K F N$ -rings and some fundamental properties of the $σ$ category, particularly when $M$ is finitely generated. Subsequently, we characterize the class of $E K F N$ -modules in rings that satisfy the ACC on annihilators.

2. Preliminary results

Lemma 1. (Theorem 3.10 of [1] ) Let $R$ be a commutative ring. These conditions are equivalent:

$R$ is an $E K F N$ -ring.
$R$ 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:

If $M$ is finitely generated as a module over $S = E n d (M)$ , then $σ = R / A n n (M)$ -Mod.
If $R$ is commutative, then for every finitely generated $R$ -module $M$ , we have $σ = R / A n n (M)$ -Mod

Proof.

For a generated set $m_{1}, m_{2}, \dots m_{k}$ of $M_{S}$ , let’s consider the map $φ : R ⟶ R (m_{1}, m_{2}, \dots m_{k}) \subset M^{k} r ⟼ r (m_{1}, m_{2}, \dots m_{k})$ We have $k e r (φ) = ⋂_{i \leq k} A n n (m_{i}) = A n n (M)$ , then $R / A n n (M) ≃ I m (φ) \subset M^{k}$ .
The second point is a consequence of $(1)$ since we have $R / A n n (M) \subset S = E n d (M)$ canonically.

Lemma 3. (From 15.2 of [2]) For two $R$ -modules $M$ , $N$ the following are equivalent:

$N$ is a subgenerator in $σ$ ;
$σ = σ [N]$ ;
$N \in σ$ and $M \in σ [N]$ .

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 $E K F N$ -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.

3. Aims results

Proposition 2. Let $M$ be an $E K F N$ -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 $E K F N$ -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 endo-noetherian. For example:

Let $R$ be the free ring over $Z$ generated by ${x_{n}, n \in N}$ . Then, $R$ is left endo-noetherian but the left ideal $I$ generated by ${x_{n}, n \in N}$ infinite direct sum of left ideals $I_{n}$ generate by ${x_{n}}$ , therefore the $R$ -module $I$ is not endo-noetherian.
$Q$ is an endo-noetherian $Z$ -module but $Q / Z = ⨁_{p \in P} Z (p^{\infty})$ is not endo-noetherian.

Proposition 3. Let $M$ be an $E K F N$ -module. Then these conditions are verified.

Every submodule of an endo-noetherian module of $σ$ is endo-noetherian.
Every quotient of an endo-noetherian module of $σ$ is endo-noetherian.

Proof. Let $M$ be an $E K F N$ -module.

Let $N$ be an endo-noetherian module in $σ$ and $P$ a submodule of $N$ . As $M$ is an $E K F N$ -module, then $N$ is noetherian and so $P$ because every submodule of a noetherian module is noetherian. Therefore $P$ is endo-noetherian.
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 $S = E n d (M)$ . If $M$ is an $E K F N$ -module then these conditions are verified:

$S = E n d (M)$ has stable range $1$ and $c o d i m (E n d (M)) \leq d i m (M) + c o d i m (M)$ .
There are, up to isomorphim, only many finitely indecomposable projective $E n d (M)$ -modules.

Proof.

$M$ finitely generated $E K F N$ -module over $S = E n d (M)$ 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 $S = E n d (M)$ is semilocal. Therefore $S = E n d (M)$ has stable range $1$ .
As $E n d (M)$ is semilocal then by theorem 4.10 of [3] we have the result.

Corollary 1. Let $M$ be a $R - E K F N$ -module. If $B$ and $C$ two arbitrary $R$ -modules and $A \oplus B ≃ A \oplus C$ , then $B ≃ C$ .

Proof. We prove this corollary by referring to Proposition 4 and Theorem 4.5 (Evans) of [3].

Proposition 5. Let $R$ be a commutative ring and $M$ a finitely generated $R$ -module. If $M$ is a $E K F N$ -module, then every object of $σ$ has a projective cover.

Proof. $M$ finitely generated over commutative ring, by referring to Lemma 2,
$σ = R / A n n (M)$ -MOD meaning that every object of $σ$ is a $R / A n n (M)$ -module. In addition, by Lemma 1, $R / A n n (M)$ is artinian principal ideal ring. So $M$ is artinian. It is well known that every artinian module is perfect. By definition of perfect ring, every object of $σ$ have projective covers.

Proposition 6. Let $R$ be a ring and $M$ a finitely generated module over $S = E n d (M)$ . Assume every simple $S$ -module has a flat cover. If $M$ is an $E K F N$ -module then $M$ is a finite direct sum of $S$ -modules with local endomorphism ring.

Proof. $M$ finitely generated over $S = E n d (M)$ implies that $M ≃ R / A n n (M)$ and $M$ $E K F N$ -module implies that $M$ is artinian. Hence $S = E n d (M)$ is semilocal. In addition, since every simple $S$ -module has a flat cover, by referring on Theorem 3.8 of [4], $S = E n d (M)$ is semiperfect. By Proposition 3.14 of [3] $M$ is a direct sum of $R$ -modules with local endomorphism ring.

Theorem 1. Let $R$ be a commuataive ring. If $M$ is a finitely generated $E K F N$ -module then the following are equivalent:

Every object of $σ$ is $c u$ -uniserial.
Every object of $σ$ is uiserial.

Proof. $(2) \Rightarrow (1)$ obvious
$(1) \Rightarrow (2)$ $M$ a finitely generated $E K F N$ -module implies $M ≃ R / A n n (M)$ is a artinian principal ideal ring. Hence $R / A n n (M)$ is a finite product of commutative artinian local rings. Suppose $R / A n n (M) = R_{1} / A n n (M) \times R_{2} / A n n (M)$ with $R_{1}$ and $R_{2}$ are local rings.
Let $K$ be a $c u$ -uniserial object of $σ$ and $L$ a non-zero finitely generated submodule of $K$ . Then $K$ is uniform and Bezout by Theorem 2.3 of [5]. This means that $L$ is cyclic and hence there exist left ideal $I_{1} = K_{1} / A n n (M)$ with $A n n (M) \subset K_{1}$ and $I_{2} = K_{2} / A n n (M)$ with $A n n (M) \subset K_{2}$ such that $L ≃ (R_{1} \times R_{2}) / I_{1} \times I_{2} ≃ (R_{1} / I_{1}) \times (R_{2} / I_{2})$ . Since $L$ is uniform, either $L ≃ (R_{1} / I_{1})$ or $L ≃ (R_{2} / I_{2})$ . Therefore $L$ is a local submodule. Hence $L / R a d (L)$ is simple submodule of $M$ thus $K$ is uniserial.

Theorem 2. Let $R$ be a ring and $M$ a finitely generated $R$ -module. We suppose $M / R a d (M)$ is a subgenerator in $σ$ . Then the following statements are equivalent.

$M$ is an $E K F N$ -module.
$M$ is a local $c u$ -uniserial module.
$M$ is a semilocal $c u$ -uniserial module.
$M$ is a virtually uniserial module.

Proof. $(1) \Rightarrow (2)$ $M$ finitely generated and $E K F N$ -module implies that $M ≃ R / A n n (M)$ is an artinian principal ideal ring. By [6], modules over commutative artinian principal ideal ring and uniseriel modules coincide. And it is easy to see that uniserial modules are $c u$ -uniserial. In addition, $M$ artinian implies $M$ is a finite product of artinian local submodules . Hence $M$ is local.
$(2) \Rightarrow (3)$ Obvious
$(1) \Leftrightarrow (4)$ It follows from theorem 2.10 of [7]
$(3) \Rightarrow (1)$ Let $N$ be an endo-noetherian object of $σ$ . $M$ semilocal, hence $M / R a d (M)$ is a semisimple artinian module. In addition $M / R a d (m)$ subgenerator in $σ$ implies $σ = σ [M / A n n (M)]$ meaning that every object of $σ [M]$ is an object of $σ [M / A n n (M)]$ . Hence $N \in σ [M / A n n (M)]$ . As $M / A n n (M)$ is semisimple then every object of $σ [M / A n n (M)]$ 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 $E K F N$ -module.

Recall $M$ 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:

$M$ is locally noetherian;
Every finitely generated module in $σ$ is noetherian;

Theorem 3. Let $M$ be a finitely generated module. The following conditions are equivalent:

$M$ is $E K F N$ -module;
$M$ is locally noetherian;

Proof. $1) ⟹ 2)$ Let $M$ an $E K F N$ -module and $N$ a finitely generated module $\in σ$ . $M$ finitely generated implies $σ$ = $R / A n n (M)$ -MOD i.e. every object of $σ$ is a $R / A n n (M)$ -module. In addition $M$ $E K F N$ -module implies by referring to Lemma 2 $M ≃ R / A n n (M)$ is an artinian and so noetherian. We know that any finitely generated module over noetherian ring is noetherian; therefore $N$ is noetherian. By Lemma 4 we can deduce that $M$ is locally noetherian.
$2) ⟹ 1)$ Let $N$ $\in σ$ an endo-noetherian module. Since $M$ is locally noetherian, by Corollary 2.3 of [8]; $R / A n n (M)$ is an noetherian ring and $σ = R / A n n (M)$ -MOD. Hence $N$ is a $R / A n n (M)$ -module. $M ≃ R / A n n (M)$ is finitely generated and noetherian. Morever $N \in σ$ implies $N$ is an ideal of $R / A n n (M)$ ; 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 $E K F N$ -module.

Corollary 2. For an $R$ -module $M$ the following assertions are equivalent:

$M$ is $E K F N$ -module;
every injective module in $σ$ is a direct sum of indecomposable. modules;

Proof. It results from Theorem 3 and 27.5 of [2].

Recall a $R$ is a $C D$ -ring if every consigular $R$ -module is discrete, and $M$ is a $C D$ -module if every $M$ -cosingular module in $σ$ is discrete.

Lemma 5. (Theorem 2.23 of [9]) The following are equivalent for a $C D$ -module $M$ .

$M$ has finite hollow dimension;
$M$ is semilocal and finitely generated

Lemma 6. (Proposition 2.26 of 1]) Let $R$ be a commutative domain. Then the following are equivalent.

$R$ is a $C D$ -ring;
Every consigular $R$ -module is projective;
$R$ is a field.

Theorem 4. Let $M$ be a $R$ -module with finite hollow dimension then the class of $E K F N$ -modules contains the class of $C D$ -modules.

Proof. Suppose that $M$ is a $C D$ -module. Let $N$ $\in σ$ an endo-noetherian module. Since $M$ is $C D$ -module with finite hollow dimension referring to Lemma 5 $M$ is finitely gererated and so $M ≃ R / A n n (M)$ . As $M$ is $C D$ -module, $R / A n n (M)$ is a $C D$ -ring and by Lemma 6 $R / A n n (M)$ is a field. Hence $R / A n n (M)$ is noetherian; $M ≃ R / A n n (M)$ is finitely generated and noetherian. Morever $N \in σ$ implies $N$ is an ideal of $R / A n n (M)$ ; 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 $E K F N$ -module.

Conflict of Interest

The authors declare no conflict of interest.

References:

Ndiaye, M.A. and Gueye, C.T., 2013. On commutative EKFN-ring with ascending chain condition on annihilators. International Journal of Pure and Applied Mathematics, 86(5), pp.871-881.[Google Scholor]
Wisbauer, R., 1991. Foundations of modules and rings theory. Gordon and Breach Science Publishers.[Google Scholor]
Facchini, A., 1998. Module theory: Endomorphism rings and direct sum decompositions in some classes of modules. Birkh\”{a}user Verlag.[Google Scholor]
Lomp, C., 1999. On semilocal modules and rings. Communications in Algebra, 27(8), pp.3961-3975.[Google Scholor]
Nikandish, R., Nikmehr, M.J. and Yassine, A., 2022. Cyclic-Uniform Uniserial Modules and Rings. arXiv preprint arXiv:2208.07940.[Google Scholor]
Faith, C., 1966. On Köthe rings. Mathematische Annalen, 164, pp.207-212.[Google Scholor]
Behboodi, M., Moradzadeh-Dehkordi, A. and Nejadi, M.Q., 2020. Virtually uniserial modules and rings. Journal of Algebra, 549, pp.365-385.[Google Scholor]
Kourki, F. and Tribak, R., 2018. Some results on locally noetherian modules and locally artinian modules. Kyungpook Mathematical Journal, 58(1), pp.1-8.[Google Scholor]
Hamzekolaee, A.R.M., Talebi, Y., Harmanci, A. and Ungor, B., 2020. Rings for which every cosingular module is discrete. Hacettepe Journal of Mathematics and Statistics, 49(5), pp.1635-1648.[Google Scholor]

[1] Ndiaye, M.A. and Gueye, C.T., 2013. On commutative EKFN-ring with ascending chain condition on annihilators. International Journal of Pure and Applied Mathematics, 86(5), pp.871-881.[Google Scholor]

[2] Wisbauer, R., 1991. Foundations of modules and rings theory. Gordon and Breach Science Publishers.[Google Scholor]

[3] Facchini, A., 1998. Module theory: Endomorphism rings and direct sum decompositions in some classes of modules. Birkh\”{a}user Verlag.[Google Scholor]

[4] Lomp, C., 1999. On semilocal modules and rings. Communications in Algebra, 27(8), pp.3961-3975.[Google Scholor]

[5] Nikandish, R., Nikmehr, M.J. and Yassine, A., 2022. Cyclic-Uniform Uniserial Modules and Rings. arXiv preprint arXiv:2208.07940.[Google Scholor]

[6] Faith, C., 1966. On Köthe rings. Mathematische Annalen, 164, pp.207-212.[Google Scholor]

[7] Behboodi, M., Moradzadeh-Dehkordi, A. and Nejadi, M.Q., 2020. Virtually uniserial modules and rings. Journal of Algebra, 549, pp.365-385.[Google Scholor]

[8] Kourki, F. and Tribak, R., 2018. Some results on locally noetherian modules and locally artinian modules. Kyungpook Mathematical Journal, 58(1), pp.1-8.[Google Scholor]

[9] Hamzekolaee, A.R.M., Talebi, Y., Harmanci, A. and Ungor, B., 2020. Rings for which every cosingular module is discrete. Hacettepe Journal of Mathematics and Statistics, 49(5), pp.1635-1648.[Google Scholor]

Contents

Utilitas Mathematica

EKFN-Modules

Abstract

1. Introduction

2. Preliminary results

3. Aims results

Conflict of Interest

References:

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