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Let RR be a commutative ring with identity and j{\mathscr{j}} an ideal of RR. An ideal II of RR is said to be a j{\mathscr{j}}-ideal if I⊈jI\hspace{0.33em} \nsubseteq \hspace{0.33em}{\mathscr{j}}. We define RR to be a j{\mathscr{j}}-Noetherian ring if each j{\mathscr{j}}-ideal of RR is finitely generated. In this work, we study some properties of j{\mathscr{j}}-Noetherian rings. More precisely, we investigate j{\mathscr{j}}-Noetherian rings via the Cohen-type theorem, the flat extension, decomposable ring, the trivial extension ring, the amalgamated duplication, the polynomial ring extension, and the power series ring extension.

Let R be a commutative ring with identity, and let S be a multiplicative subset of R. Then R is called a nonnil-S-Noetherian ring if every nonnil ideal of R is S-finite. Also, R is called a u-S-Noetherian ring if there exists an element s ∈ S such that for each ideal I of R, sI ⊆ K for some finitely generated sub-ideal K of I. In this paper, we examine some new characterization of nonnil-S-Noetherian rings. Then, as a generalization of nonnil-S-Noetherian rings and u-S-Noetherian rings, we introduce and investigate the nonnilu-S-Noetherian rings class.

In this article, we introduce two new classes of modules over a ϕ\phi -ring that generalize the classes of coherent modules and Noetherian modules. We next study the possible transfer of the properties of being nonnil-Noetherian rings, ϕ\phi -coherent rings, and nonnil-coherent rings in the amalgamated algebra along an ideal.

Let R be a commutative ring and S be a multiplicative subset of R. In this paper, we introduce and study an S-version of valuation rings, that is called the notion of S-valuation rings. Therefore, we say that R is an S-valuation ring if for all (a,b) ∈ R2, there exists an element s ∈ S such that saR ⊆ bR or sbR ⊆ aR. Moreover, we study the transfer of this property in trivial ring extensions and amalgamated algebras along an ideal. Finally, we explore the graded version of this concept in rings graded by a torsionless grading monoid. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigate two weak versions of graded-Noetherian modules. Let G be an abelian group with an identity element denoted by 0, R be a commutative G-graded ring and M be a G-graded R-module. We say that M is a graded-r-Noetherian module if every graded-r-submodule of M is finitely generated. Also, we say that M is a graded-weakly-Noetherian module if all finitely generated graded submodules of M are graded-Noetherian R-modules. We give many properties of the two different concepts and we examine the relation between them and the different concepts that already exist in the literature. We illustrate our study by giving many nontrivial examples and counter-examples. Moreover, we characterize graded-Noetherian modules in terms of graded-r-Noetherian and graded-weakly-Noetherian modules. Finally, we examine the transfer of these two concepts in the graded idealization of graded modules. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with nonzero identity. Let 𝒥(R) be the set of all ideals of R and let δ : 𝒥 (R) → 𝒥 (R) be a function. Then δ is called an expansion function of ideals of R if whenever L, I, J are ideals of R with J ⊆ I, we have L ⊆ δ (L) and δ (J) ⊆ δ (I). Let δ be an expansion function of ideals of R. In this paper, we introduce and investigate a new class of ideals that is closely related to the class of δ -primary ideals. A proper ideal I of R is said to be a 1-absorbing δ -primary ideal if whenever nonunit elements a, b, c ∈ R and abc ∈ I, then ab ∈ I or c ∈ δ (I). Moreover, we give some basic properties of this class of ideals and we study the 1-absorbing δ-primary ideals of the localization of rings, the direct product of rings and the trivial ring extensions.

The authors of A. Benhissi and S. Hizem [Nonnil-Noetherian rings and SFT property, Rocky Mountain J. Math. 41 (2011) 1483–1500] proved that the primary decomposition of a nonnil-ideal of a nonnil-Noetherian ring always exists. We will show in this paper that this primary decomposition is minimal. We next prove that over any nonnil-Noetherian ring, a sub-class of modules admit a primary decomposition. In the second part of this paper, we give some properties related to ϕ -chained and ϕ -Prüfer rings. [ABSTRACT FROM AUTHOR]

We explore the concept of the S-Krull dimension in commutative rings, extending classical notions of Krull dimension by incorporating multiplicative subsets S of a ring R. We provide characterizations of S-prime, S-maximal, and S-minimal S-prime ideals, establishing foundational results crucial for understanding the S-Krull dimension. The paper also delves into properties of S-principal ideal S-domains, S-Artinian rings, and u-S-von Neumann regular rings, introducing an S-variant of the Division Algorithm. In the final section, the concept of S-height for S-prime ideals is defined, and the S-Krull dimension is analyzed through various characterizations and examples, with a particular focus on polynomial rings. We conclude with insights into the S-height of larger S-prime ideals of R[X] and their intersections with R, contributing to a deeper understanding of the S-Krull dimension. [ABSTRACT FROM AUTHOR]

Recently, Zhang and Qi introduced and studied the concept of uniformly Sprojective (u-S-projective) modules, where S represents a multiplicative subset of a ring. In this paper, we first derive a u-S version of the well-known result that a module is projective if and only if it is a direct summand of a free module. We then provide a characterization of u-S-projective modules in terms of projective modules, specifically when S is regular, and then extend the projective basis lemma to the u-S setting. Finally, we show that a u-S-projective u-S-cover is characterized by a u-S-superfluous submodule, analogous to the way a projective cover is characterized by a superfluous submodule. [ABSTRACT FROM AUTHOR]

This paper introduces and studies the ϕ-u-S-flat (resp., nonnil-u-S-injective) modules, which are a generalization of both ϕ-flat modules and u-S-flat modules (resp., both nonnil-injective modules and u-S-injective modules). We give the Cartan–Eilenberg–Bass theorem for nonnil-u-S-Noetherian rings. Finally, we offer some new characterizations of the ϕ-von Neumann regular ring. [ABSTRACT FROM AUTHOR]

Let Γ be a torsionless grading monoid and R = ⊕ α ∈ Γ R α be a Γ -graded integral domain. In this paper, we present basic properties and give a complete characterization of strongly homogeneous prime ideals of R. These works generalize Ahmed and the authors' characterization of when R is a graded pseudo-valuation domain. [ABSTRACT FROM AUTHOR]