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In this paper, we utilize the \(\sigma\) 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.
Throughout this paper, \(R\) denotes a commutative, associative ring with an identity \(1 \ne 0\), 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 \(\sigma\) 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 \(\{ Ker(f_1) \subseteq Ker(f_2) \subseteq \cdots \}\) (or \(\{ \Im(f_1) \supseteq \Im(f_2) \supseteq \cdots \}\), respectively) stabilizes. It is immediately evident that every noetherian ring is endo-noetherian, although the converse is not universally true. For instance, \(\mathbb{Q}_{\mathbb{Z}}\) is an endo-noetherian module but not noetherian, as evidenced by the increasing sequence: \[\mathbb{Z}\subset \frac{1}{2}\mathbb{Z}\subset \frac{1}{4}\mathbb{Z}\subset\cdots\] 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 \(\sigma\) category. This new class of modules within this category is termed \(EKFN\)-modules, signifying that every endo-noetherian object in \(\sigma\) 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 \(K/\mathrm{Rad}(K)\) is both cyclic and uniform; here, \(\mathrm{Rad}(K)\) 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 \(K \subseteq M\), the quotient \(K/\mathrm{Rad}(K)\) 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 \(N+L\neq M\). 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 \(f: N \rightarrow M \rightarrow 0\) and every homomorphism \(g: P \rightarrow M\), there exists a module homomorphism \(h: P \rightarrow N\) 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 \(f: A \rightarrow B\) 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 \(f: N \rightarrow M\). \(N\) 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 \(A \rightarrow B\). 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 \(mI = 0\) for some essential right ideal \(I\) of \(R\).
We define \(\overline{Z}_M(N)\), as a dual of the singular submodule, by \(\overline{Z}_M(N) = \bigcap \{ Ker(f) ; f: N \to U, U \in \Gamma\}\), where \(\Gamma\) 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 \(EKFN\)-rings and some fundamental properties of the \(\sigma\) category, particularly when \(M\) is finitely generated. Subsequently, we characterize the class of \(EKFN\)-modules in rings that satisfy the ACC on annihilators.
Lemma 1. (Theorem 3.10 of [1] ) Let \(R\) be a commutative ring. These conditions are equivalent:
\(R\) is an \(EKFN\)-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 \(S=End(M)\), then \(\sigma=R/Ann(M)\)-Mod.
If \(R\) is commutative, then for every finitely generated \(R\)-module \(M\), we have \(\sigma=R/Ann(M)\)-Mod
Proof.
For a generated set \(m_{1}, m_{2},\cdots m_{k}\) of \(M_{S}\), let’s consider the map \[\varphi: R\longrightarrow R(m_{1}, m_{2},\cdots m_{k})\subset M^{k} \\ r\longmapsto r(m_{1}, m_{2},\cdots m_{k})\] We have \(ker(\varphi)=\bigcap_{i\leq k}{Ann(m_{i})}=Ann(M)\), then \(R/Ann(M)\simeq Im(\varphi)\subset M^{k}\).
The second point is a consequence of \((1)\) since we have \(R/Ann(M)\subset S=End(M)\) canonically.
Lemma 3. (From 15.2 of [2]) For two \(R\)-modules \(M\), \(N\) the following are equivalent:
\(N\) is a subgenerator in \(\sigma\);
\(\sigma=\sigma[N]\);
\(N\in\sigma\) and \(M\in\sigma[N]\).
Recall a module \(M\) is a \(S\)-module if every hopfian object of \(\sigma\) 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 \(\sigma\). \(N\) 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.
Proposition 2. Let \(M\) be an \(EKFN\)-module. Then the homomorphic image of every endo-noetherian module of \(\sigma\) is endo-noetherian.
Proof. Let \(N\) be an endo-noetherian module in \(\sigma\); as \(M\) is an \(EKFN\)-module, then \(N\) is noetherian. Assume that \(f: N\longrightarrow f(N)=K\) 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 \(\mathbf{Z}\) 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.
\(\mathbb{Q}\) is an endo-noetherian \(\mathbb{Z}\)-module but \(\mathbb{Q}/\mathbb{Z} = \bigoplus_{p\in\mathbb{P}}\mathbf{Z}(p^{\infty})\) is not endo-noetherian.
Proposition 3. Let \(M\) be an \(EKFN\)-module. Then these conditions are verified.
Every submodule of an endo-noetherian module of \(\sigma\) is endo-noetherian.
Every quotient of an endo-noetherian module of \(\sigma\) is endo-noetherian.
Proof. Let \(M\) be an \(EKFN\)-module.
Let \(N\) be an endo-noetherian module in \(\sigma\) 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.
If \(N\) is an endo-noetherian module of \(\sigma\) 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:
\(S=End(M)\) has stable range \(1\) and \(codim(End(M))\leq dim(M)+codim(M)\).
There are, up to isomorphim, only many finitely indecomposable projective \(End(M)\)-modules.
Proof.
\(M\) 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 \(1\).
As \(End(M)\) is semilocal then by theorem 4.10 of [3] we have the result.
Corollary 1. Let \(M\) be a \(R-EKFN\)-module. If \(B\) and \(C\) two arbitrary \(R\)-modules and \(A\oplus B\simeq A\oplus C\), then \(B\simeq C\).
Proposition 5. Let \(R\) be a commutative ring and \(M\) a finitely generated \(R\)-module. If \(M\) is a \(EKFN\)-module, then every object of \(\sigma\) has a projective cover.
Proof. \(M\) finitely
generated over commutative ring, by referring to Lemma 2,
\(\sigma=R/Ann(M)\)-MOD meaning that
every object of \(\sigma\) is a \(R/Ann(M)\)-module. In addition, by Lemma 1,
\(R/Ann(M)\) 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 \(\sigma\) have projective covers.
Proposition 6. Let \(R\) be a ring and \(M\) a finitely generated module over \(S=End(M)\). Assume every simple \(S\)-module has a flat cover. If \(M\) is an \(EKFN\)-module then \(M\) is a finite direct sum of \(S\)-modules with local endomorphism ring.
Proof. \(M\) finitely generated over \(S=End(M)\) implies that \(M\simeq R/Ann(M)\) and \(M\) \(EKFN\)-module implies that \(M\) is artinian. Hence \(S=End(M)\) is semilocal. In addition, since every simple \(S\)-module has a flat cover, by referring on Theorem 3.8 of [4], \(S=End(M)\) 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 \(EKFN\)-module then the following are equivalent:
Every object of \(\sigma\) is \(cu\)-uniserial.
Every object of \(\sigma\) is uiserial.
Proof. \((2)\Rightarrow
(1)\) obvious
\((1)\Rightarrow (2)\) \(M\) a finitely generated \(EKFN\)-module implies \(M\simeq R/Ann(M)\) is a artinian principal
ideal ring. Hence \(R/Ann(M)\) is a
finite product of commutative artinian local rings. Suppose \(R/Ann(M)= R_{1}/Ann(M)\times R_{2}/Ann(M)\)
with \(R_{1}\) and \(R_{2}\) are local rings.
Let \(K\) be a \(cu\)-uniserial object of \(\sigma\) 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 \(I_{1}=K_{1}/Ann(M)\) with \(Ann(M)\subset K_{1}\) and \(I_{2}=K_{2}/Ann(M)\) with \(Ann(M)\subset K_{2}\) such that \(L\simeq (R_{1}\times R_{2})/I_{1}\times I_{2}
\simeq (R_{1}/I_{1})\times (R_{2}/I_{2})\). Since \(L\) is uniform, either \(L\simeq (R_{1}/I_{1})\) or \(L\simeq(R_{2}/I_{2})\). Therefore \(L\) is a local submodule. Hence \(L/Rad(L)\) 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 \(M/Rad(M)\) is a subgenerator in \(\sigma\). Then the following statements are equivalent.
\(M\) is an \(EKFN\)-module.
\(M\) is a local \(cu\)-uniserial module.
\(M\) is a semilocal \(cu\)-uniserial module.
\(M\) is a virtually uniserial module.
Proof. \((1)\Rightarrow
(2)\) \(M\) finitely generated
and \(EKFN\)-module implies that \(M\simeq R/Ann(M)\) 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 \(cu\)-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 \(\sigma\). \(M\) semilocal, hence \(M/Rad(M)\) is a semisimple artinian module.
In addition \(M/Rad(m)\) subgenerator
in \(\sigma\) implies \(\sigma=\sigma[M/Ann(M)]\) meaning that
every object of \(\sigma[M]\) is an
object of \(\sigma[M/Ann(M)]\). Hence
\(N\in \sigma[M/Ann(M)]\). As \(M/Ann(M)\) is semisimple then every object
of \(\sigma[M/Ann(M)]\) 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.
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 \(\sigma\) is noetherian;
Theorem 3. Let \(M\) be a finitely generated module. The following conditions are equivalent:
\(M\) is \(EKFN\)-module;
\(M\) is locally noetherian;
Proof. \(1) \implies 2)\)
Let \(M\) an \(EKFN\)-module and \(N\) a finitely generated module \(\in \sigma\). \(M\) finitely generated implies \(\sigma\) = \(R/Ann(M)\)-MOD i.e. every object of \(\sigma\) is a \(R/Ann(M)\) -module. In addition \(M\) \(EKFN\)-module implies by referring to Lemma
2
\(M \simeq R/Ann(M)\) 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.
\(2) \implies 1)\) Let \(N\) \(\in
\sigma\) an endo-noetherian module. Since \(M\) is locally noetherian, by Corollary 2.3
of [8]; \(R/Ann(M)\) is an noetherian ring and \(\sigma = R/Ann(M)\)-MOD. Hence \(N\) is a \(R/Ann(M)\)-module. \(M \simeq R/Ann(M)\) is finitely generated
and noetherian. Morever \(N \in
\sigma\) 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.
Corollary 2. For an \(R\)-module \(M\) the following assertions are equivalent:
\(M\) is \(EKFN\)-module;
every injective module in \(\sigma\) is a direct sum of indecomposable. modules;
Recall a \(R\) is a \(CD\)-ring if every consigular \(R\)-module is discrete, and \(M\) is a \(CD\)-module if every \(M\)-cosingular module in \(\sigma\) is discrete.
Lemma 5. (Theorem 2.23 of [9]) The following are equivalent for a \(CD\)-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 \(CD\)-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 \(EKFN\)-modules contains the class of \(CD\)-modules.
Proof. Suppose that \(M\) is a \(CD\)-module. Let \(N\) \(\in \sigma\) an endo-noetherian module. Since \(M\) is \(CD\)-module with finite hollow dimension referring to Lemma 5 \(M\) is finitely gererated and so \(M \simeq R/Ann(M)\). As \(M\) is \(CD\)-module, \(R/Ann(M)\) is a \(CD\)-ring and by Lemma 6 \(R/Ann(M)\) is a field. Hence \(R/Ann(M)\) is noetherian; \(M \simeq R/Ann(M)\) is finitely generated and noetherian. Morever \(N \in \sigma\) 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.
The authors declare no conflict of interest.