On n-absorbing rings and ideals

Abdallah Laradji

doi:10.4064/cm6844-5-2016

On n-absorbing rings and ideals

Abdallah Laradji^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

14 Scopus citations

Abstract

A proper ideal I of a commutative ring R is n-absorbing (resp. strongly n-absorbing) if for all elements (resp. ideals) a₁,., a_n+1of R/I, a₁· · · a_n+1= 0 implies that the product of some n of the ai is 0. It was conjectured by Anderson and Badawi that if I is an n-absorbing ideal of R then (1) I is strongly n-absorbing, (2) I[x] is an n-absorbing ideal of R[x], and (3) Rad(I)ⁿ⊆ I. We prove that these conjectures hold in various classes of rings, thus extending several known results on n-absorbing ideals. As a by-product, we show that (2) implies (1).

Original language	English
Pages (from-to)	265-274
Number of pages	10
Journal	Colloquium Mathematicum
Volume	147
Issue number	2
DOIs	https://doi.org/10.4064/cm6844-5-2016
State	Published - 2017

Bibliographical note

Publisher Copyright:
© Instytut Matematyczny PAN, 2017.

Keywords

N-absorbing ideal
N-ring
Prüfer ring
Strongly n-absorbing ideal

ASJC Scopus subject areas

General Mathematics

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10.4064/cm6844-5-2016

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AU - Laradji, Abdallah

N1 - Publisher Copyright: © Instytut Matematyczny PAN, 2017.

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AB - A proper ideal I of a commutative ring R is n-absorbing (resp. strongly n-absorbing) if for all elements (resp. ideals) a1,., an+1of R/I, a1· · · an+1= 0 implies that the product of some n of the ai is 0. It was conjectured by Anderson and Badawi that if I is an n-absorbing ideal of R then (1) I is strongly n-absorbing, (2) I[x] is an n-absorbing ideal of R[x], and (3) Rad(I)n⊆ I. We prove that these conjectures hold in various classes of rings, thus extending several known results on n-absorbing ideals. As a by-product, we show that (2) implies (1).

KW - N-absorbing ideal

KW - N-ring

KW - Prüfer ring

KW - Strongly n-absorbing ideal

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DO - 10.4064/cm6844-5-2016

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JO - Colloquium Mathematicum

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On n-absorbing rings and ideals

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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