On $SCDF$-Modules

Diompy, Mankagna; A, Alhousseynou; Diabang, Andé

doi:/10.61091/um118-06

Abstract

1. Introduction
2. Some properties of SCDF-modules
3. Characterization theorems
Conflict of Interest

References

Utilitas Mathematica

Volume 118

Research article

On $S C D F$ -Modules

^¹, ^¹, ^¹

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

Received: 19/09/2023
Accepted: 27/12/2023
Published: 08/01/2024

Copyright Link
License

Abstract

A module $M$ over a commutative ring is termed an $S C D F$ -module if every Dedekind finite object in $σ [M]$ is finitely cogenerated. Utilizing this concept, we explore several properties and characterize various types of $S C D F$ -modules. These include local $S C D F$ -modules, finitely generated $SCDF$-modules, and hollow $S C D F$ -modules with $R a d (M) = 0 \neq M$ . Additionally, we examine $Q F$ SCDF-modules in the context of duo-ri

Keywords: Dedekind finite, Finitely cogenerated,

S C D F

-modules, Hollow modules,

Q F

-modules

1. Introduction

Throughout this paper, all rings are assumed to be either commutative or duo-rings with $1 \neq 0$ . A module $M$ over a commutative ring is termed Dedekind finite (respectively, finitely cogenerated) if every monomorphism $f : M ⟶ M$ is an automorphism (respectively, if $S o c (M)$ is essential and finitely generated). Although any finitely cogenerated module is Dedekind finite, the converse is not generally true. In light of this fact, we utilize the $σ [M]$ category to introduce the concept of an $S C D F$ -module, which is a generalization of the $S C D F$ -ring. We define a non-zero $R$ -module $M$ as hollow if every proper submodule of it is superfluous. The socle of a module $M$ , denoted as $S o c (M)$ , is defined as the sum of its minimal non-zero submodules. Conversely, the radical of $M$ is the intersection of all its maximal submodules. Let $C$ be a subcategory of $R$ -Mod. A module $N$ in $C$ is called finitely presented (for shot f.p) in $C$ if:

$N$ is finitely generated and
Every exact sequence $0 \to K \to L \to N \to 0$ in $C$ , with $L$ finitely generated, $K$ is also finitely generated.

Let $M$ be an $R$ -module. A module $N \in σ [M]$ is called coherent in $σ [M]$ if

$N$ is finitely generated and
any finitely generated submodule of $N$ is finitely presented in $σ [M]$ .

A non-zero module $M$ is called a hollow module if every proper submodule of $M$ is a small submodule of $M$ . Let $M$ be a faithful $R$ -module. We say that $M$ is a Quasi-Frobenius (in short QF) module if $H o m_{R} (P, M)$ is either zero or a simple $R$ -module for each simple $R$ -module $P$ .

2. Some properties of $S C D F$ -modules

Proposition 1.

Epimorphic image of $S C D F$ -module is a $S C D F$ -module;
If $M$ is a product of modules $M_{i}$ , $1 \leq i \leq n$ is a $S C D F$ -module. Then so is every $M_{i}$ ;
Moreover if $H o m (M_{i}, M_{j}) = 0$ for all $1 \leq i \neq j \leq n$ , then the converse of $(2)$ is true;
Every factor module of $S C D F$ -module is a $S C D F$ -module.

Proof.

Let $M$ be a $S C D F$ -module and $M^{^{'}} = f (M)$ a homomorphic image of $M$ , then $G e n (M^{^{'}})$ is in $G e n (M)$ [1]. This implies that $σ [M^{^{'}}]$ is a full subcategory of $σ [M]$ . Hence $M^{^{'}}$ is a $S C D F$ -module of finite.
Results from $(1)$ .
Suppose that every $M_{i}$ for $1 \leq i \leq n$ is a $S C D F$ -module. As $H o m (M_{i}, M_{j}) = 0$ for $M_{i}$ for $1 \leq i \neq j \leq n$ , then by Proposition 2.2 of [2], for every $N \in σ [\prod_{i = 1}^{n} M_{i}]$ there is a unique $N_{i} \in σ [M_{i}]$ $1 \leq i \leq n$ such that $N = \prod_{i = 1}^{n} N_{i}$ . If $N$ is a Dedekind finite, $N_{i}$ is also a Dedekind finite for all $1 \leq i \leq n$ because if a module is a Dedekind finite, then so is any direct summand of that module. Since $M_{i}$ is a $S C D F$ -module, then $N_{i}$ is finitely cogenerated for all $1 \leq i \leq n$ . Hence $N = \prod_{i = 1}^{n} N_{i}$ is finitely cogenerated. Thus, $M$ is a $S C D F$ -module.
Let $M$ a $S C D F$ -module and $N$ a submodule of $M$ . Then by proposition $(15.1)$ of [3], $M / N \in σ [M]$ . Let $K \in σ [M / N]$ , $K$ is also belongs $σ [M]$ . Wich implies $K$ is finitely cogenerated. Therefore $M / N$ is a $S C D F$ -module.

Lemma 1. (15.4 of [3])

If a $R$ -module $M$ is finitely generated as a module over $S = E n d_{R} (M)$
then $σ [M] = R / A n n (M)$ -MOD.
If $R$ is commutative, then for every finitely generated $R$ -module we have $σ [M] = R / A n n (M)$ -MOD.

Lemma 2. (Theorem 2.1 of [4])
Let $R$ be a commutative ring. Then the following statements are equivalent:

$R$ is an artinian principal ideal ring
$R$ is a $S C D F$ -ring.

Proposition 2. Let $M$ a local $R$ -module over $S = E n d (M)$ . If $M$ is $S C D F$ -module, then $M$ is coherent.

Proof. If $M$ is local over $S$ , then $M$ is finitely generated over $S = E n d (M)$ . Referring to the first point of Lemma 1 and Lemma 2 $M ≅ R / A n n (M)$ and $R / A n n (M)$ is artinian. It is well know that any artinian ring is noetherian. Therefore $M$ coherent because every noetherian module is coherent.

Recall an $R$ -module $M$ is called locally artinian if every finitely generated module in $σ [M]$ is artinian.

Lemma 3. (41.4 of [3])
Let $M$ be a non-zero $R$ -module. The following are equivalent:

$M$ is hollow module and $R a d (M) \neq M$
$M$ is local.

Proposition 3. Let $R$ commutative ring and $M$ a hollow module with $R a d (M) = 0 \neq M$ . If $M$ is a $S C D F$ -module, then $M$ is locally artinian.

Proof. Let $N$ be a finitely generated module in $σ [M]$ . Successively by Lemmas 3, 1 and 2, $M$ is artinian. In addition, $R a d (M) = 0$ implies by referring to [1] proposition $10.15$ $M$ is semisimple and noetherian. If $M$ is semisimple, every module of $σ [M]$ is also semisimple. By [1] Corollary $10.16$ finitely generated and artinian are equivalent. Therefore $M$ is locally artinian.

3. Characterization theorems

Lemma 4. Suppose a ring $R$ has zero nilradical. $R$ is Dedekind finite if and only if $R$ is artinian.

Definition 1. A non empty set $S$ of a ring $R$ is said to be a mutiplicatively closed subset (briefly m.c.s) if $1 \in S$ and $a b \in S$ for each $a, b \in S$ .

Remark 1. We denote the set of all prime and maximal ideals of $R$ by $S p e c (R)$ and $M a x (R)$ respectively.

A ring $R$ is called a $S$ -artinian if for each descending chain of ideals ${I_{i}}_{i \in N}$ of $R$ these exist $s \in S$ and $k \in N$ such that $s I_{k} \subseteq I_{n}$ for all $n \geq k$ .

Lemma 5 (By example 3 of [5]). Every artinian $R$ -module $M$ is a $S$ -artinian module where $S \subseteq R$ is m.c.s.

Definition 2. Let $M$ be a $R$ -module. A propoer submodule $P$ of $M$ is said to be prime if for any $r \in R$ and $m \in M$ with $r m \in P$ , we have $m \in P$ or $r \in (P :_{R} M)$ .
$M$ is said to be reduced if intersection of all prime submodules of $M$ is equal to zero.

If $N$ is a submodule of a $R$ -module, then $c l (N) = {m \in M, m I \subseteq N for some largeleft ideal of R}$ . If $c l (N) = N$ , then $N$ is said to be closed.

Definition 3. A submodule $N$ of a $R$ -module $M$ is termed closed prime provided the following two conditions are satisfied:

if $N^{'}$ is a submodule such that $N \subset N^{'} \subseteq M$ , then $(N : N^{'}) \subset (N : M)$ .
$c l (N) = M$

Theorem 1. Let $M$ be a finitely generated module over commutative ring. The following properties are equivalent.

$M$ is a $S C D F$ -module
$M$ is artinian
$M$ is $P$ -artinian for each $P \in S p e c (R)$
$M$ is $μ$ -artinian for each $μ \in M a x (R)$ .

Proof. $(1) \Rightarrow (2)$ Result from Proposition 1
$(2) \Leftrightarrow (3) \Leftrightarrow (4)$ these equivalences follow the theorem $2$ [5]
$(2) \Rightarrow (1)$ Let $N$ be a Dedekind finite object of $σ [M]$ . There is an epimorphism $φ : M^{(Λ)} ⟶ K$ such that $N$ is a submodule of $K$ . Since $M$ is finitely generated then $c a r d (Λ)$ is finite. The first theorem of isomorphism implies $M^{(Λ)} / k e r φ ≃ K$ .
$M$ artinian, $c a r d (Λ)$ finite implies $M^{(Λ)}$ is also artinian. Since artinian modules are stable of submodules and factor modules, then $K$ and $N$ are artinian. Since $N$ is artinian, has a simple submodule, in fact $S o c (N)$ is an essential submodule.
Now let’s prouve that $S o c (N)$ is finitely generated.
$M$ finitely generated implies $σ [M] ≃ R / A n n (M)$ -MOD. Hence every object of $σ [M]$ is a $R / A n n (M)$ -module therefore $R / A n n (M)$ is an artinian ring. In addition for an artinian ring, finitely generated is equivalent to artinian. Then $N$ is finitely generated. $S o c (N)$ is also finitely generated because every submodule of artinian finitely generated module is finitely generated.

Theorem 2. Let $M$ be an hollow module and $R a d (M) \neq M$ . The following are equivalent:

$M$ is a $S C D F$ -module
$M$ is locally artinian
Every cyclic module in $σ [M]$ is a direct sum of a self-projective $M$ -injective module and a finitely cogenerated module.

Proof. $(1) \Rightarrow (2)$ Result of Proposition 3
$(2) \Rightarrow (1)$ $M$ locally artinian and $M$ finitely generated implies $M$ artinian. In addition, $R a d (M) = 0$ implies $M$ is semisimple. Then $M = ⨁_{i} M_{i}$ is a direct sum of a finite set of simple modules. $N \in σ [M]$ , then $N$ is also semisimple and finite lenght. Therefore $N$ is artinian and noetherian. For a semisimple module, artinian, noetherian and finitely cogenerated coincide.
$(2) \Leftrightarrow (3)$ Results from 3.Theorem of [6].

Recall a $R$ -module $M$ is uniform if every non-zero submodule of $M$ is an essential submodule.

Definition 4. Let $M$ be a module such that every module in $σ [M]$ is a direct sum of uniform modules. Then we will say that $M$ is an $S U$ -module.

Definition 5. A module $M$ is said to be pure-semisimple if every module in $σ [M]$ is a direct sum of finitely presented modules.

Theorem 3. Let $R$ be a commutative ring and $M$ a local $R$ -module. Then the following are equivalent:

$M$ is a $S C D F$ -module;
$M$ is a $S U$ -module;
$M$ is pure-semisimple;

Proof. Referring Lemmas 1 and 2 $M ≃ R / A n n (M)$ is an artinian ideal principal ring. Basing on [7], artinian with principal ideal and uniserial coincide. Therefore $M$ is uniserial. In addition, it is well known that every uniserial module is serial. Basing on Theorem 5.2.1 from [8] we have the equivalences.

Definition 6. A uniserial module $M$ is said to be homo-uniserial if whenever $A$ , $B$ , $C$ and $D$ are submodules of $M$ such that $A$ and $C$ are maximal submodules of $B$ and $D$ respectively, then $B / A ≃ D / C$ .

Corollary 1. Let $M$ be a module over a commutative ring $R$ . Then the following are equivalent:

$M$ is a $S C D F$ -module;
Every module in $σ [M]$ is a direct sum of homo-uniserial modules.
Every module in $σ [M]$ is a direct sum of homo-uniserial modules of finite length.

Proof. Results from Theorem 3 and Theorem 5.2.4 of [8]

Now, we suppose that $R$ is a duo-ring. It is a ring such that every one-sided ideal is two-sided ideal.We have the following theorem.

Theorem 4. Let $R$ be a duo-ring and $M$ faithfully balanced left finite generated $R$ -module. Assume $s o c (M)$ is square-free. then the following statements are equivalent

$M$ is a $S C D F$ -module;
$M$ is a $Q F$ -module;
$M ≃ M_{1} \times M_{2} \times \dots \dots \times M_{s}$ where each $M_{i}$ is a local artinian module with a simple socle.

Proof. Assume $S = E n d (M)$ the ring of endomorphism of $M$ . If $M$ is an $R$ -module, then $M$ is an $S$ -module. Thus it follows from the first point of lemma 1 $σ [M] = R / A n n (M)$ -Mod. As $R / A n n (M)$ is a duo ring, and M is isomorphic to $R / A n n (M)$ and from Theorem $9$ of [9] $M$ is artinian .Since $M$ is also square-free,it results from Theorem $15.27$ of [10] that the statements are equivalent.

Corollary 2. Let $R$ be a duo-ring and M faithfully balanced left finite generated $R$ -module. Assume $s o c (M)$ is square-free. then the following statements are equivalent

$M$ is a $S C D F$ -module;
$M$ is a $Q F$ -module;
$M$ is self injective;
$M$ is a cogenerator;

Proof. The corollary results from Theorem 4 and theorem 30.7 from [1]

Conflict of Interest

The authors declare no conflict of interest.

References:

Vanaja, N., 1996. All finitely generated $M$ -subgenerated modules are extending. Communications in Algebra, 24(2), pp.543-572.[Google Scholor]
Wisbauer, R., 1991. Foundations of module and ring theory. Gordon and Breach Science Publishers.[Google Scholor]
Labou, I., Toure, S.D. and Sanghare, M., 2013. A characterisation of commutative rings in which every Dedekind finite module is finitely cogenerated. African Mathematical Annals, AFMA, 4, pp.87-89.
Anderson, F.W. and Fuller, K., 1974. Rings and categories of modules. Springer-Verlag.[Google Scholor]
Ozen, M., Nazi, O.A., Tekir, U. and Shum, K.P., 2021. Characterization theorems of $S$ -Artinian modules. Comptes rendus de I’Academie bulgare des Sciences, 74(4), pp.496-505.[Google Scholor]
Wisbauer, R. and Huynh, D.V., 1990. A characterization of locally Artinian modules. Journal of Algebra, 132, pp.287-293.[Google Scholor]
Faith, C., (1966). On Kothe rings. Mathematische Annalen, 164, pp.207-212.[Google Scholor]
Loggie, William T.H., 1997. Some special classes of modules. Doctoral thesis, University of Glasgow.[Google Scholor]
Kaidi, A.M. and Sangharé, M., 1988. Une caractérisation des anneaux artiniens à idéaux principaux. Lecture Notes in Mathematics, Springer Verlag, pp.245-254.
Lam, T.Y., 1991. Lectures on modules and rings. Springer.[Google Scholor]

[1] Vanaja, N., 1996. All finitely generated $M$ -subgenerated modules are extending. Communications in Algebra, 24(2), pp.543-572.[Google Scholor]

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

[3] Labou, I., Toure, S.D. and Sanghare, M., 2013. A characterisation of commutative rings in which every Dedekind finite module is finitely cogenerated. African Mathematical Annals, AFMA, 4, pp.87-89.

[4] Anderson, F.W. and Fuller, K., 1974. Rings and categories of modules. Springer-Verlag.[Google Scholor]

[5] Ozen, M., Nazi, O.A., Tekir, U. and Shum, K.P., 2021. Characterization theorems of $S$ -Artinian modules. Comptes rendus de I’Academie bulgare des Sciences, 74(4), pp.496-505.[Google Scholor]

[6] Wisbauer, R. and Huynh, D.V., 1990. A characterization of locally Artinian modules. Journal of Algebra, 132, pp.287-293.[Google Scholor]

[7] Faith, C., (1966). On Kothe rings. Mathematische Annalen, 164, pp.207-212.[Google Scholor]

[8] Loggie, William T.H., 1997. Some special classes of modules. Doctoral thesis, University of Glasgow.[Google Scholor]

[8] Kaidi, A.M. and Sangharé, M., 1988. Une caractérisation des anneaux artiniens à idéaux principaux. Lecture Notes in Mathematics, Springer Verlag, pp.245-254.

[10] Lam, T.Y., 1991. Lectures on modules and rings. Springer.[Google Scholor]

Contents

Utilitas Mathematica

On $S C D F$ -Modules

Abstract

1. Introduction

2. Some properties of $S C D F$ -modules

3. Characterization theorems

Conflict of Interest

References:

Information

Guidelines

CP Initiatives

Follow CP

Contents

Utilitas Mathematica

On SCDF-Modules

Abstract

1. Introduction

2. Some properties of SCDF-modules

3. Characterization theorems

Conflict of Interest

References:

Information

Guidelines

CP Initiatives

Follow CP

On $S C D F$ -Modules

2. Some properties of $S C D F$ -modules