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

<p>Let <italic>R</italic> be a commutative ring with nonzero identity. Let 𝒥(<italic>R</italic>) be the set of all ideals of <italic>R</italic> and let <italic>δ</italic> : 𝒥 (<italic>R</italic>) → 𝒥 (<italic>R</italic>) be a function. Then <italic>δ</italic> is called an expansion function of ideals of <italic>R</italic> if whenever <italic>L, I, J</italic> are ideals of <italic>R</italic> with <italic>J</italic> ⊆ <italic>I</italic>, we have <italic>L</italic> ⊆ <italic>δ</italic> (<italic>L</italic>) and <italic>δ</italic> (<italic>J</italic>) ⊆ <italic>δ</italic> (<italic>I</italic>). Let <italic>δ</italic> be an expansion function of ideals of <italic>R</italic>. In this paper, we introduce and investigate a new class of ideals that is closely related to the class of <italic>δ</italic> -primary ideals. A proper ideal <italic>I</italic> of <italic>R</italic> is said to be a 1-absorbing <italic>δ</italic> -primary ideal if whenever nonunit elements <italic>a, b, c</italic> ∈ <italic>R</italic> and <italic>abc</italic> ∈ <italic>I</italic>, then <italic>ab</italic> ∈ <italic>I</italic> or <italic>c</italic> ∈ <italic>δ</italic> (<italic>I</italic>). Moreover, we give some basic properties of this class of ideals and we study the 1-absorbing <italic>δ</italic>-primary ideals of the localization of rings, the direct product of rings and the trivial ring extensions.</p>
</abstract>

Let R be a commutative ring with nonzero identity. Let 𝒥(R) be the set of all ideals of R and let δ : 𝒥 (R) → 𝒥 (R) be a function. Then δ is called an expansion function of ideals of R if whenever L, I, J are ideals of R with J ⊆ I, we have L ⊆ δ (L) and δ (J) ⊆ δ (I). Let δ be an expansion function of ideals of R. In this paper, we introduce and investigate a new class of ideals that is closely related to the class of δ -primary ideals. A proper ideal I of R is said to be a 1-absorbing δ -primary ideal if whenever nonunit elements a, b, c ∈ R and abc ∈ I, then ab ∈ I or c ∈ δ (I). Moreover, we give some basic properties of this class of ideals and we study the 1-absorbing δ-primary ideals of the localization of rings, the direct product of rings and the trivial ring extensions.

On 1-absorbing <italic>δ</italic>-primary ideals

Laboratory of Modelling and Mathematical Structures

Department of Mathematic, Faculty of Science and Technology of Fez

Department of Mathematics, Faculty of Science and Technology of Fez, Box 2202

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

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

{"article-title":{"italic":"δ","content":["On 1-absorbing","-primary ideals"]}}

Let R be a commutative ring with nonzero identity. Let 𝒥(R) be the set of all ideals of R and let δ : 𝒥 (R) → 𝒥 (R) be a function. Then δ is called an...

On 1-absorbing δ-primary ideals

Pubblicato online: 23 nov 2021

Pagine: 135 - 150

Ricevuto: 09 mar 2021

Accettato: 30 apr 2021

DOI: https://doi.org/10.2478/auom-2021-0038

Parole chiave
1-absorbing prime ideal, 1-absorbing -primary ideal, -primary ideal, trivial ring extension

© 2021 Abdelhaq El Khalfi et al., published by Sciendo

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

On 1-absorbing δ-primary ideals

Pubblicato online: 23 nov 2021

Pagine: 135 - 150

Ricevuto: 09 mar 2021

Accettato: 30 apr 2021

DOI: https://doi.org/10.2478/auom-2021-0038

Parole chiave1-absorbing prime ideal, 1-absorbing -primary ideal, -primary ideal, trivial ring extension

© 2021 Abdelhaq El Khalfi et al., published by Sciendo

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

Parole chiave
1-absorbing prime ideal, 1-absorbing -primary ideal, -primary ideal, trivial ring extension