The Zariski topology on the graded second spectrum of a graded module

Salam, Saif; Al-Zoubi, Khaldoun

doi:10.21203/rs.3.rs-1335019/v1

Cited by 3 publications

(3 citation statements)

References 6 publications

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“…The following example shows that the reverse inclusion in Proposition 3.6(3) is not true in general. [16,Lemma 2.8(1)], every non-zero graded submodule of a graded module over a graded field is graded second, which follows that…”

Section: Graded Modules With Noetherian Graded Second Spectrummentioning

confidence: 99%

“…When N does not contain graded second submodules, we set soc G (N ) = {0}. For more information about the graded second submodules and the graded second socle of graded submodules of graded modules (see, for example, [5,8,16]).…”

Section: Introductionmentioning

confidence: 99%

“…For more details concerning the topologies on Spec s G (M ) and the natural map of Spec s G (M ), one can look in [16]. Let M be a G-graded R-module.…”

Section: Introductionmentioning

confidence: 99%

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Graded modules with Noetherian graded second spectrum

Salam¹,

Al-Zoubi²

2022

Preprint

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Let R be a G graded commutative ring and M be a Ggraded R-module. The set of all graded second submodules of M is denoted by Spec s G (M ) and it is called the graded second spectrum of M . In this paper, we discuss graded rings with Noetherian graded prime spectrum and obtain some conclusions. In addition, we introduce the notion of the graded Zariski socle of graded submodules and explore their properties. Using these conclusions and properties, we also investigate Spec s G (M ) with the Zariski topology from the viewpoint of being a Noetherian space and give some related outcomes.

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Section: Graded Modules With Noetherian Graded Second Spectrummentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Graded modules with Noetherian graded second spectrum

Salam¹,

Al-Zoubi²

2022

Preprint

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Zariski topology on the secondary-like spectrum of a module

Salam,

Al-Zoubi

2024

Open Mathematics

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Let ℜ \Re be a commutative ring with unity and ℑ \Im be a left ℜ \Re -module. We define the secondary-like spectrum of ℑ \Im to be the set of all secondary submodules K K of ℑ \Im such that the annihilator of the socle of K K is the radical of the annihilator of K K , and we denote it by Spec L ( ℑ ) {{\rm{Spec}}}^{L}\left(\Im ) . In this study, we introduce a topology on Spec L ( ℑ ) {{\rm{Spec}}}^{L}\left(\Im ) having the Zariski topology on the second spectrum Spec s ( ℑ ) {{\rm{Spec}}}^{s}\left(\Im ) as a subspace topology and study several topological structures of this topology.

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Graded modules with Noetherian graded second spectrum

Salam¹,

Al-Zoubi

2023

MATH

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<abstract><p>Let $ R $ be a $ G $ graded commutative ring and $ M $ be a $ G $-graded $ R $-module. The set of all graded second submodules of $ M $ is denoted by $ Spec_G^s(M), $ and it is called the graded second spectrum of $ M $. We discuss graded rings with Noetherian graded prime spectrum. In addition, we introduce the notion of the graded Zariski socle of graded submodules and explore their properties. We also investigate $ Spec^s_G(M) $ with the Zariski topology from the viewpoint of being a Noetherian space.</p></abstract>

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The Zariski topology on the graded second spectrum of a graded module

Cited by 3 publications

References 6 publications

Graded modules with Noetherian graded second spectrum

Graded modules with Noetherian graded second spectrum

Zariski topology on the secondary-like spectrum of a module

Graded modules with Noetherian graded second spectrum

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