On $SCDF$-Modules

Diompy, Mankagna; A, Alhousseynou; Diabang, Andé

doi:/10.61091/um118-06

Abstract

References

Utilitas Mathematica

Volume 118

Research article

On $SCDF$-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

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Abstract

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

Keywords: Dedekind finite, Finitely cogenerated, $SCDF$-modules, Hollow modules, $QF$-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 $f: M \longrightarrow M$ is an automorphism (respectively, if $Soc(M)$ 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 $\sigma[M]$ category to introduce the concept of an $SCDF$-module, which is a generalization of the $SCDF$-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 $Soc(M)$, 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 $\mathcal{C}$ be a subcategory of $R$-Mod. A module $N$ in $\mathcal{C}$ is called finitely presented (for shot f.p) in $\mathcal{C}$ if:

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

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

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

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 $Hom_{R}(P,M)$ is either zero or a simple $R$-module for each simple $R$-module $P$.

2. Some properties of $SCDF$-modules

Proposition 1.

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

Proof.

Let $M$ be a $SCDF$-module and $M^{'}= f(M)$ a homomorphic image of $M$, then $Gen(M^{'})$ is in $Gen(M)$ [1]. This implies that $\sigma[M^{'}]$ is a full subcategory of $\sigma[M]$. Hence $M^{'}$ is a $SCDF$-module of finite.
Results from $(1)$.
Suppose that every $M_{i}$ for $1\leq i \leq n$ is a $SCDF$-module. As $Hom(M_{i},M_{j} )=0$ for $M_{i}$ for $1\leq i\neq j \leq n$, then by Proposition 2.2 of [2], for every $N\in \sigma[\prod_{i=1}^{n}M_{i}]$ there is a unique $N_{i}\in \sigma [M_{i}]$ $1\leq i \leq n$ 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 $SCDF$-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 $SCDF$-module.
Let $M$ a $SCDF$-module and $N$ a submodule of $M$. Then by proposition $(15.1)$ of [3], $M/N \in \sigma[M]$. Let $K\in \sigma[M/N]$, $K$ is also belongs $\sigma[M]$. Wich implies $K$ is finitely cogenerated. Therefore $M/N$ is a $SCDF$-module.

Lemma 1. (15.4 of [3])

If a $R$-module $M$ is finitely generated as a module over $S=End_{R}(M)$
then $\sigma[M]=R/Ann(M)$-MOD.
If $R$ is commutative, then for every finitely generated $R$-module we have $\sigma[M]=R/Ann(M)$-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 $SCDF$-ring.

Proposition 2. Let $M$ a local $R$-module over $S = End(M)$. If $M$ is $SCDF$-module, then $M$ is coherent.

Proof. If $M$ is local over $S$, then $M$ is finitely generated over $S = End(M)$. Referring to the first point of Lemma 1 and Lemma 2 $M \cong R/Ann(M)$ and $R/Ann(M)$ 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 $\sigma[M]$ 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 $Rad(M)\neq M$
$M$ is local.

Proposition 3. Let $R$ commutative ring and $M$ a hollow module with $Rad(M)=0\neq M$. If $M$ is a $SCDF$-module, then $M$ is locally artinian.

Proof. Let $N$ be a finitely generated module in $\sigma[M]$. Successively by Lemmas 3, 1 and 2, $M$ is artinian. In addition, $Rad(M)=0$ implies by referring to [1] proposition $10.15$ $M$ is semisimple and noetherian. If $M$ is semisimple, every module of $\sigma[M]$ 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 $ab\in S$ for each $a, b\in S$.

Remark 1. We denote the set of all prime and maximal ideals of $R$ by $Spec(R)$ and $Max(R)$ respectively.

A ring $R$ is called a $S$-artinian if for each descending chain of ideals $\{I_{i}\}_{i\in \mathbf{N}}$ of $R$ these exist $s\in S$ and $k\in \mathbf{N}$ such that $sI_{k}\subseteq I_{n}$ 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 $rm\in P$, 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 $cl(N)=\{m\in M, mI\subseteq N \text{for some large left ideal of } R\}$. If $cl(N)=N$, 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 $(N: N')\subset (N: M)$.
$cl(N)=M$

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

$M$ is a $SCDF$-module
$M$ is artinian
$M$ is $P$-artinian for each $P\in Spec(R)$
$M$ is $\mu$-artinian for each $\mu\in Max(R)$.

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 $\sigma[M]$. There is an epimorphism $\varphi : M^{(\Lambda)}\longrightarrow K$ such that $N$ is a submodule of $K$. Since $M$ is finitely generated then $card(\Lambda)$ is finite. The first theorem of isomorphism implies $M^{(\Lambda)}/ker\varphi\simeq K$.
$M$ artinian, $card(\Lambda)$ finite implies $M^{(\Lambda)}$ 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 $Soc(N)$ is an essential submodule.
Now let’s prouve that $Soc(N)$ is finitely generated.
$M$ finitely generated implies $\sigma[M]\simeq R/Ann(M)$-MOD. Hence every object of $\sigma[M]$ is a $R/Ann(M)$-module therefore $R/Ann(M)$ is an artinian ring. In addition for an artinian ring, finitely generated is equivalent to artinian. Then $N$ is finitely generated. $Soc(N)$ is also finitely generated because every submodule of artinian finitely generated module is finitely generated.

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

$M$ is a $SCDF$-module
$M$ is locally artinian
Every cyclic module in $\sigma[M]$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, $Rad(M)=0$ implies $M$ is semisimple. Then $M=\bigoplus_{i}M_{i}$ is a direct sum of a finite set of simple modules. $N\in\sigma[M]$, 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 $\sigma[M]$ is a direct sum of uniform modules. Then we will say that $M$ is an $SU$-module.

Definition 5. A module $M$ is said to be pure-semisimple if every module in $\sigma[M]$ 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 $SCDF$-module;
$M$ is a $SU$-module;
$M$ is pure-semisimple;

Proof. Referring Lemmas 1 and 2 $M\simeq R/Ann(M)$ 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 $B/A\simeq D/C$.

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

$M$ is a $SCDF$-module;
Every module in $\sigma[M]$ is a direct sum of homo-uniserial modules.
Every module in $\sigma[M]$ 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 $soc(M)$ is square-free. then the following statements are equivalent

$M$ is a $SCDF$-module;
$M$ is a $QF$-module;
$M\simeq M_{1}\times M_{2}\times……\times M_{s}$ where each $M_{i}$ is a local artinian module with a simple socle.

Proof. Assume $S = End(M)$ 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 $\sigma [M]= R/Ann(M)$-Mod. As $R/Ann(M)$ is a duo ring, and M is isomorphic to $R/Ann(M)$ 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 $soc(M)$ is square-free. then the following statements are equivalent

$M$ is a $SCDF$-module;
$M$ is a $QF$-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 \(SCDF\)-Modules

Abstract

1. Introduction

2. Some properties of \(SCDF\)-modules

3. Characterization theorems

Conflict of Interest

References:

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