This article introduces the concept of finitely null-additive set function relative to the σ– ring and many properties of this concept have been discussed. Furthermore, to introduce and study the notion of finitely weakly null-additive set function relative to the σ– ring as a generalization of some concepts such as measure, countably additive, finitely additive, countably null-additive, countably weakly null-additive and finitely null-additive. As the first result, it has been proved that every finitely null-additive is a finitely weakly null-additive. Finally, the paper introduces a study of the concept of outer measure as a stronger form of finitely weakly null-additive.

The concept of the Extend Nearly Pseudo Quasi-2-Absorbing submodules was recently introduced by Omar A. Abdullah and Haibat K. Mohammadali in 2022, where he studies this concept and it is relationship to previous generalizationsm especially  2-Absorbing submodule and Quasi-2-Absorbing submodule, in addition to studying the most important Propositions, charactarizations and Examples. Now in this research, which is considered a continuation of the definition that was presented earlier, which is the Extend Nearly Pseudo Quasi-2-Absorbing submodules, we have completed the study of this concept in multiplication modules. And the relationship between the Extend Nearly Pseudo Quasi-2-Absorbing submodule and Extend Nearly Pseudo Quasi-2-Abs

   Let R be a commutative ring with unity and let M be a unitary R-module. In this paper we study fully semiprime submodules and fully semiprime modules, where a proper fully invariant R-submodule W of M is called fully semiprime in M if whenever XXW for all fully invariant R-submodule X of M, implies XW.         M is called fully semiprime if (0) is a fully semiprime submodule of M. We give basic properties of these concepts. Also we study the relationships between fully semiprime submodules (modules) and other related submodules (modules) respectively.

The duo module plays an important role in the module theory. Many researchers generalized this concept such as Ozcan AC, Hadi IMA and Ahmed MA. It is known that in a duo module, every submodule is fully invariant. This paper used the class of St-closed submodules to work out a module with the feature that all St-closed submodules are fully invariant. Such a module is called an Stc-duo module. This class of modules contains the duo module properly as well as the CL-duo module which was introduced by Ahmed MA. The behaviour of this new kind of module was considered and studied in detail,for instance, the hereditary property of the St-duo module was investigated, as the result; under certain conditions, every St-cl

Let  be a module over a commutative ring  with identity. Before studying the concept of the Strongly Pseudo Nearly Semi-2-Absorbing submodule, we need to mention the ideal  and the basics that you need to study the  concept of the Strongly Pseudo Nearly Semi-2-Absorbing submodule. Also, we introduce several characteristics of the Strongly Pseudo Nearly Semi-2-Absorbing submodule in classes of multiplication modules and other types of modules. We also had no luck because the ideal  is not a Strongly Pseudo Nearly Semi-2-Absorbing ideal. Also, it is noted that  is the Strongly Pseudo Nearly Semi-2-Absorbing ideal under several conditions, which is this faithful module, projective module, Z-regular module and content module and non-si

The concept of a 2-Absorbing submodule is considered as an essential feature in the field of module theory and has many generalizations. This articale discusses the concept of the Extend Nearly Pseudo Quasi-2-Absorbing submodules and their relationship to the 2-Absorbing submodule, Quasi-2-Absorbing submodule, Nearly-2-Absorbing submodule, Pseudo-2-Absorbing submodule, and the rest of the other concepts previously studied. The relationship between them has been studied, explaining that the opposite is not true and that under certain conditions the opposite becomes true. This article aims to study this concept and gives the most important propositions, characterizations, remarks, examples, lemmas, and observations related to it. In the en

    We introduce in this paper the concept of approximaitly semi-prime submodules of unitary left -module  over a commutative ring  with identity as a generalization of a prime submodules and semi-prime submodules, also generalization of quasi-prime submodules and approximaitly prime submodules. Various basic properties of an approximaitly semi-prime submodules are discussed, where a proper submodule  of an -module  is called an approximaitly semi-prime submodule of  , if whenever , where ,  and , implies that . Furthermore the behaviors of approximaitly semi-prime submodule in some classes of modules are studied. On the other hand several characterizations of this concept are

Let R be a commutative ring with identity 1 and M be a unitary left R-module. A submodule N of an R-module M is said to be pure relative to submodule T of M (Simply T-pure) if for each ideal A of R, N?AM=AN+T?(N?AM). In this paper, the properties of the following concepts were studied: Pure essential submodules relative to submodule T of M (Simply T-pure essential),Pure closed submodules relative to submodule T of M (Simply T-pure closed) and relative pure complement submodule relative to submodule T of M (Simply T-pure complement) and T-purely extending. We prove that; Let M be a T-purely extending module and let N be a T-pure submodule of M. If M has the T-PIP, then N is T-purely extending.

  Let L be a commutative ring with identity and let W be a unitary left L- module. A submodule D of an L- module W is called  s- closed submodule denoted by  D ≤sc W, if D has   no  proper s- essential extension in W, that is , whenever D ≤ W such that D ≤se H≤ W, then D = H. In  this  paper,  we study  modules which satisfies  the ascending chain  conditions (ACC) and descending chain conditions (DCC) on this kind of submodules.