Let M be a module over a domain R and M # = { 0 ≠ m ∈ M : Rm ≠ M } be the set of all nonzero nongenerators of M. Consider the following equivalence relation ∼ on M # given by m ∼ n if and only if Rm = Rn for every m , n ∈ M # . Let EC ( M # ) be the set of all equivalence classes of M # with respect to ∼ . In this paper, we construct a topology on EC ( M # ) which is called the divisor topology of M and is denoted by D ( M ) . Actually, D ( M ) is an extension of the divisor topology D ( R ) over domains to modules in the sense of Yiğit and Koç. We investigate separation axioms T i for every 0 ≤ i ≤ 5 , first and second countability, connectivity, compactness, nested property, and Noetherian property on D ( M ) . Also, we characterize some important classes of modules such as uniserial modules, simple modules, vector spaces, and finitely cogenerated modules in terms of D ( M ) . Furthermore, we prove that D ( M ) is a Baire space for factorial modules. Finally, we introduce and study pseudo simple modules which is a new generalization of simple modules, and use them to determine when D ( M ) is a discrete space.

In this paper, we study weakly uniserial modules, a concept recently introduced by Moradzadeh-Dehkordi et al., which extends the notion of uniserial modules. A module $M$ is said to be weakly uniserial if for any submodules $N$ and $L$ of $M$, there exists a monomorphism $N \rightarrowtail L$ or $L \rightarrowtail N$. Our analysis explores the relationship between weakly uniserial modules and classical notions in ring and module theory, including preradicals, socle series, the singular submodule, injective hulls, and $V$-rings. In addition, we present further statements complementing the characterization provided by the aforementioned authors concerning rings over which every module is weakly uniserial. Finally, by using monomorphisms, we resolve the Schröder–Bernstein problem within the class of isoartinian modules.

Given the significance of abelian $p$-groups in module theory and their connection to algebraic structures, this paper focuses on constructing the $QTAG$-modules using the notion of torsion abelian groups and investigating their algebraic counterparts. These include examining specific types of submodules, such as isotype submodules, high submodules, and $h$-pure submodules, as well as exploring the concept of subsocles. In addition, we analyze the relationship between the concept of summability and the direct sum of uniserial modules within the context of these $QTAG$-modules.

Let R be a commutative unital ring and N be a submodule of an R-module M. We show that: 1) the semiprime radical is an invariant on submodules generated by the ascending chain of envelopes of a given submodule; 2) for rings that satisfy the radical formula, ⟨EM (0)⟩ is an idempotent radical leading to a torsion theory whose torsion class has nil R-modules and the torsion-free class has reduced R-modules; and, 3) Noetherian uniserial modules satisfy the semiprime radical formula and their semiprime radical is a nil module.

A uniserial module is a module that satisfies both ascending chain condition and descending chain condition, which makes a uniserial module an Artinian module and a Noetherian module at the same time. Recently an algebraic structure from ring theory, called almost prime ideal, is generalized into a module theory and called an almost prime submodule. Some researchers have examined the characterizations of this new algebraic structure in various types of modules. In this article, we provide some insights into the almost prime submodule of a uniserial cyclic module In this study, we have discovered that the non-zero almost prime submodule of the cyclic uniserial module is unique.

A module is called weakly uniserial if for any two its submodules at least one of them is embedded in the other. This is a nontrivial generalization of uniserial modules and rings. Here, we introduce and study the dual of this concept. In fact, an R ‐module M is called coweakly uniserial if for any submodules N , K of M , Hom R ( M / N , M / K ) or Hom R ( M / K , M / N ) contains a surjective element. In this paper, in addition to presenting the properties of this concept, we show that a ring R is homogeneous semisimple if and only if every (projective) right R ‐module is coweakly uniserial. Also, in a semi‐Artinian ring R , it is shown that if every 2‐generated right R ‐module is coweakly uniserial, then R is a homogeneous semisimple ring. Then, we prove that has no coweakly uniserial subgroups. Among applications of our results, we classify quasi‐continuous, quasi‐injective, and uniform abelian groups that are coweakly uniserial.

The string almost gentle algebras (SAG-algebras) are studied in this paper. Generalizing the properties of string almost gentle algebras has also given. Let be the string almost gentle algebras, with quiver and admissible ideal of algebra. We show that the radical of string almost gentle algebras can be written as a direct sum of uniserial modules. After that the quiver is constructed from and an extension of the quiver is showed. Also, the quiver which is a quiver of trivial extension of string almost gentle algebras has been studied. Consequently, the relations of algebra have been described. As well as, we will show that the algebra is an extension to string almost gentle algebras. Furthermore, we describe the trivial extension of string almost gentle algebras , we prove and that the trivial extension of is isomorphic to the algebra .

A right module M over an associative ring R with unity is a QT AG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. We show the clo- sures properties for certain high submodules by the QT AG-modules and vice versa. Important generalizations and certain related assertions of classical results in this direction are also established.

In this paper, we study a generalization of $h$-pure submodules as well as some other closely related concepts. Here, we examine the extent of this generalization in several ways. We then use this to give a characterization of the imbedded-complete modules. It is found that imbeddedness can considerably more abundant than $h$-purity on direct sum of uniserial modules.

Let R be an associative ring and M a unitary left R-module. An R-module M is said to be uniserial if its submodules are linearly ordered by inclusion. A serial module is a direct sum of uniserial modules. In this paper, we bring our modest contribution to the open problem listed in the book of Alberto Facchini "Module Theory" which states that is any direct summand of a serial module serial? The answer is yes for particular rings and R-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.

The paper is concerned with h-pure-N-high submodules of QTAG-modules. Here, we characterize the submodules N of an h-reduced QTAG-module for which all h-pure-N-high submodules are bounded. We also discuss some interesting properties of subsocles and consequently give a characterization of the direct sum of uniserial modules.

Direct sum decompositions of projective and injective modules into virtually uniserial modules

A right module <em>M</em> over an associative ring <em>R</em> with unity is a <em>QTAG</em>-module if every finitely generated submodule of any homomorphic image of <em>M</em> is a direct sum of uniserial modules. Here we characterize the finitely generated submodule <em>N</em> of a <em>QTAG</em>-module <em>M</em> such that all homomorphisms or monomorphisms of the finitely generated submodule <em>N</em> into the <em>QTAG</em>-module <em>M</em>, or all endomorphisms of the finitely generated submodule <em>N</em>, extends to an endomorphism of the <em>QTAG</em>-module <em>M</em>.

The module is a generalization of the vector space. The module that will be discussed is a uniserial module, which is a module that only has one composition series. A uniserial ring is a ring whose module over itself is uniserial. The uniserial ring is an Artin and local ring, but the converse is not necessarily true. In this paper, we will discuss Artin and local ring with additional properties so that it is characteristics of the uniserial ring.

Suppose M is a QTAG-module with a subsocle S such that M/S is a direct sum of uniserial modules. Our global aim here is to investigate an interesting connection between the structure of M/S and the QTAG-module M. Specifically, the condition S=Soc(N) for some h-pure submodules N of M allows M to inherit the structure of M/S.

A module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. In this paper, we study the existence of several classes C of QTAG-modules which satisfy the property that M belongs to C uniquely when M/N belongs to C provided that N is a finitely generated submodule of the QTAG-module.

Let R be a ring, K,M be R-modules, L a uniserial R-module, and X a submodule of L. The triple (K,L,M) is said to be X-sub-exact at L if the sequence K→X→M is exact. Let σ(K,L,M) is a set of all submodules Y of L such that (K,L,M) is Y -sub-exact. The sub-exact sequence is a generalization of an exact sequence. We collect all triple (K,L,M) such that (K,L,M) is an X-sub exact sequence, where X is a maximal element of σ(K,L,M). In a uniserial module, all submodules can be compared under inclusion. So, we can find the maximal element of σ(K,L,M). In this paper, we prove that the set σ(K,L,M) form a category, and we denoted it by C_L. Furthermore, we prove that C_Y is a full subcategory of C_L, for every submodule Y of L. Next, we show that if L is a uniserial module, then C_L is a pre-additive category. Every morphism in C_L has kernel under some conditions. Since a module factor of L is not a submodule of L, every morphism in a category C_L does not have a cokernel. So, C_L is not an abelian category. Moreover, we investigate a monic X-sub-exact and an epic X-sub-exact sequence. We prove that the triple (K,L,M) is a monic X-sub-exact if and only if the triple Z-modules (<img src=image/13424409_01.gif>, <img src=image/13424409_02.gif>, <img src=image/13424409_03.gif>) is a monic <img src=image/13424409_04.gif>-sub-exact sequence, for all R-modules N. Furthermore, the triple (K,L,M) is an epic X-sub-exact if and only if the triple Z-modules (<img src=image/13424409_05.gif>, <img src=image/13424409_06.gif>, <img src=image/13424409_07.gif>) is a monic <img src=image/13424409_08.gif>-sub-exact, for all R-module N.

Nilpotency degree of the nilradical of a solvable Lie algebra on two generators and uniserial modules associated to free nilpotent Lie algebras

A module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of universal modules. In this paper, we investigate the class of QTAG-modules having nice basis. It is proved that if H_ω (M) is bounded then M has a bounded nice basis and if H_ω (M) is a direct sum of uniserial modules, then M has a nice basis. We also proved that if M is any QTAG-module, then M⊕D has a nice basis, where D is the h-divisible hull of H_ω (M).