Chain conditions on composite Hurwitz series rings

Lim, Jung Wook; Oh, Dong Yeol

doi:10.1515/math-2017-0097

Cited by 10 publications

(3 citation statements)

References 15 publications

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“…Thereafter S-Noetherian rings and modules were continuously studied by many authors (see [4], [6], [10], [11], [12], [13], [14], [15] and [16], for example). This notion has motivated many researchers to study S-version of known structures in ring and module theory (see [4], [6], [21] and [22], for example).…”

Section: Introductionmentioning

confidence: 99%

Graded S-Noetherian Modules

ANSARI

Sharma

2023

International Electronic Journal of Algebra

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Let $G$ be an abelian group and $S$ a given multiplicatively closed subset of a commutative $G$-graded ring $A$ consisting of homogeneous elements. In this paper, we introduce and study $G$-graded $S$-Noetherian modules which are a generalization of $S$-Noetherian modules. We characterize $G$-graded $S$-Noetherian modules in terms of $S$-Noetherian modules. For instance, a $G$-graded $A$-module $M$ is $G$-graded $S$-Noetherian if and only if $M$ is $S$-Noetherian, provided $G$ is finitely generated and $S$ is countable. Also, we generalize some results on $G$-graded Noetherian rings and modules to $G$-graded $S$-Noetherian rings and modules.

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Section: Introductionmentioning

confidence: 99%

Graded S-Noetherian Modules

ANSARI

Sharma

2023

International Electronic Journal of Algebra

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“…Furthermore, if S is the set of units, then any nonnil-Noetherian ring that is not a Noetherian ring is not an S-Noetherian ring. The readers can refer to [1, 3,4] for nonnil-Noetherian rings and to [2,[5][6][7][8][9][10][11] for S-Noetherian rings.…”

Section: Introductionmentioning

confidence: 99%

On Nonnil-S-Noetherian Rings

Kwon

Lim

2020

Mathematics

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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.

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“…Furthermore, we say that M is S-finite if sM ⊆ F for some s ∈ S and some finitely generated R-submodule F of M; and M is S-Noetherian if every R-submodule of M is S-finite. For more on S-Noetherian rings and S-finiteness, the readers can refer to [3][4][5][6][7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

A Note on Weakly S-Noetherian Rings

Kim

Lim

2020

Symmetry

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Let R be a commutative ring with identity and S a (not necessarily saturated) multiplicative subset of R. We call the ring R to be a weakly S-Noetherian ring if every S-finite proper ideal of R is an S-Noetherian R-module. In this article, we study some properties of weakly S-Noetherian rings. In particular, we give some conditions for the Nagata’s idealization and the amalgamated algebra to be weakly S-Noetherian rings.

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Chain conditions on composite Hurwitz series rings

Cited by 10 publications

References 15 publications

Graded S-Noetherian Modules

Graded S-Noetherian Modules

On Nonnil-S-Noetherian Rings

A Note on Weakly S-Noetherian Rings

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