<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>Let <italic>R</italic> be a commutative ring with identity and <italic>S</italic> be a multiplicative subset of <italic>R</italic>. In this paper, we introduce the concept of weakly <italic>S</italic>-prime ideals which is a generalization of weakly prime ideals. Let <italic>P</italic> be an ideal of <italic>R</italic> disjoint with <italic>S</italic>. We say that <italic>P</italic> is a weakly <italic>S</italic>-prime ideal of <italic>R</italic> if there exists an <italic>s</italic> ∈ <italic>S</italic> such that, for all <italic>a, b</italic> ∈ <italic>R</italic>, if 0 ≠ <italic>ab</italic> ∈ <italic>P</italic>, then <italic>sa</italic> ∈ <italic>P</italic> or <italic>sb</italic> ∈ <italic>P</italic>. We show that weakly <italic>S</italic>-prime ideals have many analog properties to these of weakly prime ideals. We also use this new class of ideals to characterize <italic>S</italic>-Noetherian rings and <italic>S</italic>-principal ideal rings.</p>
</abstract>

Let R be a commutative ring with identity and S be a multiplicative subset of R. In this paper, we introduce the concept of weakly S-prime ideals which is a generalization of weakly prime ideals. Let P be an ideal of R disjoint with S. We say that P is a weakly S-prime ideal of R if there exists an s ∈ S such that, for all a, b ∈ R, if 0 ≠ ab ∈ P, then sa ∈ P or sb ∈ P. We show that weakly S-prime ideals have many analog properties to these of weakly prime ideals. We also use this new class of ideals to characterize S-Noetherian rings and S-principal ideal rings.

On weakly <italic>S</italic>-prime ideals of commutative rings

Department of Mathematics, Faculty of Science

Team of Modeling and Scientific Computing, Mathematics Department, Pluridisciplinary faculty

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

On weakly -prime ideals of commutative rings

{"article-title":{"italic":"S","content":["On weakly","-prime ideals of commutative rings"]}}

Let R be a commutative ring with identity and S be a multiplicative subset of R. In this paper, we introduce the concept of weakly S-prime ideals which is a...

On weakly S-prime ideals of commutative rings

Published Online: Jul 08, 2021

Page range: 173 - 186

Received: Dec 11, 2020

Accepted: Jan 19, 2021

DOI: https://doi.org/10.2478/auom-2021-0024

Keywords
-prime ideals, Weakly -prime ideals, -Noetherian rings, Nagata idealization

© 2021 Fuad Ali Ahmed Almahdi et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

On weakly S-prime ideals of commutative rings

Published Online: Jul 08, 2021

Page range: 173 - 186

Received: Dec 11, 2020

Accepted: Jan 19, 2021

DOI: https://doi.org/10.2478/auom-2021-0024

Keywords-prime ideals, Weakly -prime ideals, -Noetherian rings, Nagata idealization

© 2021 Fuad Ali Ahmed Almahdi et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
-prime ideals, Weakly -prime ideals, -Noetherian rings, Nagata idealization