S-Noetherian Rings of the Forms 𝒜[X] and 𝒜[[X]]

Hamed, Ahmed; Hizem, Sana

doi:10.1080/00927872.2014.924127

Cited by 23 publications

(7 citation statements)

References 5 publications

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“…On the other hand, the notion of (commutative) S-Noetherian rings was introduced by Anderson and Dumitrescu who proved the S-variant of Eakin-Nagata theorem on commutative rings in 2002 [4, Corollary 7]. Then it was further studied in [1,3,14,[17][18][19]25]. While the research of noncommutative S-Noetherian rings is started recently by a few authors, they found some valuable results.…”

Section: Introductionmentioning

confidence: 99%

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Eakin–Nagata–Eisenbud Theorem for Right $S$-Noetherian Rings

2023

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The Eakin-Nagata theorem examines the condition that the Noetherian property passes through each other between subrings and extension rings in 1968. Later, a noncommutative version of Eakin-Nagata theorem was also proved. Lam called this version Eakin-Nagata-Eisenbud theorem. In addition, Anderson and Dumitrescu introduced the S-Noetherian concept which is an extended notion of the Noetherian property on commutative rings in 2002. In this paper, we consider the S-variant of Eakin-Nagata-Eisenbud theorem for general rings by using S-Noetherian modules. We also show that every right S-Noetherian domain is right Ore, which is embedded into a division ring. For a right S-Noetherian ring, we obtain sufficient conditions for its right ring of fractions to be right S-Noetherian or right Noetherian. As applications, the S-variant of Eakin-Nagata-Eisenbud theorem is applied to the composite polynomial, composite power series and composite skew polynomial rings.

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Section: Introductionmentioning

confidence: 99%

“…(1) Let D be a Noetherian integral domain, T = D \{0}, X = {X i | i ∈ N} a set of indeterminates over D and R…”

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confidence: 99%

Eakin–Nagata–Eisenbud Theorem for Right $S$-Noetherian Rings

2023

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“…In this work R is a commutative ring with identity, I is an ideal and S is a multiplicative (closed under multiplication) subset of R whose elements are regular. Many classic concepts from ideal theory are generalized to S-concepts for instance see [1], [2], [3], [4], [5]. In [6], Anderson et al defined the ideal I to be an S-finite ideal, if there exists an element s of S and a finitely generated ideal J satisfying: sI ⊆ J ⊆ I.…”

Section: Introductionmentioning

confidence: 99%

S-Strongly Finite Type Rings

Eljeri

2018

ARJOM

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In this paper a ring means a commutative ring with identity element and S is a multiplicatively closed subset of the studied ring whose elements are regular. Inspiring from the work of J. Arnold about strongly finite type rings and D. E. Rush , s about noetherian spectrum rings, we introduce two types of rings that are S-strongly finite type rings and S-noetherian spectrum rings. We give some characterizations of these rings and we illustrate them by many examples.

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“…They succeeded to generalize several well-known results on Noetherian rings including the classical Cohen's result and Hilbert basis theorem under an additional condition. Since then the S-finiteness has attracted the interest of several authors (see for instance [6,7,10,11,12,14]). Recentely, motivated by the work of Anderson and Dumitrescu, S-versions of some classical notions have been introduced (see for instance [6,10]).…”

Section: Introductionmentioning

confidence: 99%

On S-coherence

Bennis¹,

Hajoui²

2016

Preprint

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Recentely, Anderson and Dumitrescu's S-finiteness has attracted the interest of several authors. In this paper, we introduce the notions of Sfinitely presented modules and then of S-coherent rings which are S-versions of finitely presented modules and coherent rings, respectively. Among other results, we give an S-version of the classical Chase's characterization of coherent rings. We end the paper with a brief discussion on other S-versions of finitely presented modules and coherent rings. We prove that these last S-versions can be characterized in terms of localization.

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S-Noetherian Rings of the Forms 𝒜[X] and 𝒜[[X]]

Cited by 23 publications

References 5 publications

Eakin–Nagata–Eisenbud Theorem for Right $S$-Noetherian Rings

Eakin–Nagata–Eisenbud Theorem for Right $S$-Noetherian Rings

S-Strongly Finite Type Rings

On S-coherence

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