Let \(R=\bigoplus _{\alpha \in \Gamma } R_{\alpha }\) be a commutative ring graded by any arbitrary torsionless grading monoid \(\Gamma \). In this article, we introduce the graded version of the notion of Prüfer rings, which is a generalization of graded Prüfer domains to the context of arbitrary \(\Gamma \)-graded rings with zero-divisors and we extend some properties to the graded situation.

In 2002, D.D. Anderson and T. Dumitrescu introduced S-Noetherian rings, a significant advancement in ring theory. Later, in 2018, D. Bennis and M. El Hajoui expanded on this by introducing two new classes of rings that generalize the concept of coherent rings: S-coherent rings, where every finitely generated ideal is S-finitely presented, and c-S-coherent rings, where every S-finite ideal is S-finitely presented. This paper builds on these concepts to introduce the c-S-(weak) global dimension for rings with a multiplicative subset S, offering a new perspective on ring structures. We also explore c-S-variants of specific ring types, including semisimple, von Neumann, and (semi)hereditary rings. Finally, we extend the generalization of (n, d)-rings and n-coherent rings, originally introduced by D.L. Costa in 1994, to rings with a multiplicative subset S, deepening the understanding of ring theory.

Let [Formula: see text] be a commutative ring and [Formula: see text] be a multiplicative subset of [Formula: see text]. In this paper, we introduce and study an [Formula: see text]-version of valuation rings, that is called the notion of [Formula: see text]-valuation rings. Therefore, we say that [Formula: see text] is an [Formula: see text]-valuation ring if for all [Formula: see text], there exists an element [Formula: see text] such that [Formula: see text] or [Formula: see text]. 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.

In this paper, we extend the concepts of Noetherian and coherent modules to ℵ_0-Noetherian and ℵ_0-coherent modules. A module is defined as ℵ_0-Noetherian if every submodule is countably generated, and ℵ_0-coherent if it is countably generated and every countably generated submodule is countably presented. We investigate the fundamental properties of these modules and explore their behavior under several constructions, such as trivial ring extensions, fiber products, and amalgamations. Additionally, we demonstrate that ℵ_0-Noetherian and ℵ_0-coherent modules are closed under localization and provide new insights into the structure of these modules in comparison to their classical counterparts.

In this paper, we introduce and investigate two weak versions of graded-Noetherian modules. Let [Formula: see text] be an abelian group with an identity element denoted by [Formula: see text], [Formula: see text] be a commutative [Formula: see text]-graded ring and [Formula: see text] be a [Formula: see text]-graded [Formula: see text]-module. We say that [Formula: see text] is a graded-r-Noetherian module if every graded-r-submodule of [Formula: see text] is finitely generated. Also, we say that [Formula: see text] is a graded-weakly-Noetherian module if all finitely generated graded submodules of [Formula: see text] are graded-Noetherian [Formula: see text]-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.