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In this paper, we introduce a new class of ring called w p - S -Noetherian ring, which is a weak version of S -Noetherian ring property and study the transfer of this notion to various context of commutative ring extensions such as direct product, trivial ring extensions and amalgamation of rings. Furthermore, we define the concept of nonnil w p - S -Noetherian ring property which is a generalization of the w p - S -Noetherian domain property and establish a characterization of this notion using pullbacks. [ABSTRACT FROM AUTHOR]

Let A be a commutative integral domain with identity, S a multiplicatively closed subset of A. An ideal I of A is called S-principal if there exist s ∈ S and x ∈ I such that sI ⊆< x >⊆ I. Also, an S-integral domain A is called S-PID if every ideal I of A is Sprincipal. Using this concept, we give many examples, properties and S-versions of several known results on principal ideal domains. Also, we characterize S-PID in terms of S-prime ideals. Moreover, we investigate structural properties of modules over S-PID. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of ring called regular S-Noetherian ring, which is a weak version of S-Noetherian ring property. Any S-Noetherian ring is naturally a regular S-Noetherian ring, and in the domain context, these two forms coincide. We study the transfer of this notion to various context of commutative ring extensions such as direct product, trivial ring extensions and amalgamated duplication of a ring along an ideal. Our results generate new families of examples of non-S-Noetherian regular S-Noetherian rings. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, M an R-module and S ⊆ R a multiplicative set. Then M is called S-finite if there exist an s ∈ S and a finitely generated submodule N of M such that sM ⊆ N. Also, M is called S-Noetherian if each submodule of M is S-finite. A ring R is called S-Noetherian if it is S-Noetherian as an R-module. This paper surveys the most recent developments in describing the structural properties of S-Noetherian rings, S-Noetherian modules and their generalizations. Some interesting constructed examples of S-Noetherian rings and modules are also presented.

Let R be a commutative ring with identity, M an R-module and S ⊆ R a multiplicative set. Then M is called S-finite if there exist an s ∈ S and a finitely generated submodule N of M such that sM ⊆ N. Also, M is called S-Noetherian if each submodule of M is S-finite. A ring R is called S-Noetherian if it is S-Noetherian as an R-module. This paper surveys the most recent developments in describing the structural properties of S-Noetherian rings, S-Noetherian modules and their generalizations. Some interesting constructed examples of S-Noetherian rings and modules are also presented. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and S a multiplicative subset of R. In this paper, the notion of S -pseudo-irreducible ideals of R as a generalization of pseudo-irreducible ideals is introduced and some related properties are investigated. Then, we consider the decomposition of proper ideals of a ring into a product of pairwise comaximal S -pseudo-irreducible ideals. Some other results are obtained too. [ABSTRACT FROM AUTHOR]

In this paper, we study chain conditions on composite Hurwitz series rings and composite Hurwitz polynomial rings. More precisely, we characterize when composite Hurwitz series rings and composite Hurwitz polynomial rings are Noetherian, S-Noetherian or satisfy the ascending chain condition on principal ideals.

Let Γ be a commutative monoid, R=⨁α∈ΓRα a Γ-graded ring and S a multiplicative subset of R0. We define R to be a graded S-Noetherian ring if every homogeneous ideal of R is S-finite. In this paper, we characterize when the ring R is a graded S-Noetherian ring. As a special case, we also determine when the semigroup ring is a graded S-Noetherian ring. Finally, we give an example of a graded S-Noetherian ring which is not an S-Noetherian ring.

Let R be a commutative ring with identity, and let S be a (not necessarily saturated) multiplicative subset of R. We define R to be a nonnil-S-Noetherian ring if each nonnil ideal of R is S-finite. In this paper, we study some properties of nonnil-S-Noetherian rings. More precisely, we investigate nonnil-S-Noetherian rings via the Cohen-type theorem, the flat extension, the faithfully flat extension, the polynomial ring extension, and the power series ring extension.

This paper studies Hurwitz series rings and Hurwitz polynomial rings. Especially we characterize nilpotent elements, units, McCoy theorem, Noetherian, nonnil-Noetherian and S-Noetherian properties, PS and presimplifiable properties. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and S ⊆ R a given multiplicative set. Let (M, ≤) be a strictly ordered monoid satisfying the condition that 0 ≤ m for every m ∈ M. Then it is shown, under some additional conditions, that the generalized power series ring [[RM' ≤]] is S-Noetherian if and only if R is S-Noetherian and M is finitely generated. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, and let S be a (not necessarily saturated) multiplicative subset of R. We define R to be a nonnil-S-Noetherian ring if each nonnil ideal of R is S-finite. In this paper, we study some properties of nonnil-S-Noetherian rings. More precisely, we investigate nonnil-S-Noetherian rings via the Cohen-type theorem, the flat extension, the faithfully flat extension, the polynomial ring extension, and the power series ring extension. [ABSTRACT FROM AUTHOR]