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On $n$-absorbing rings and ideals

Volume 147 / 2017

Abdallah Laradji Colloquium Mathematicum 147 (2017), 265-273 MSC: Primary 13A15; Secondary 13F05. DOI: 10.4064/cm6844-5-2016 Published online: 23 January 2017

Abstract

A proper ideal $I$ of a commutative ring $R$ is $n$-absorbing (resp. strongly $n$-absorbing) if for all elements (resp. ideals) $a_{1},\ldots ,a_{n+1}$ of $R/I$, $a_{1}\cdots a_{n+1}=0$ implies that the product of some $n$ of the $a_{i}$ 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) $\mathrm {Rad}(I)^{n}\subseteq I$. 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).

Authors

  • Abdallah LaradjiDepartment of Mathematics & Statistics
    King Fahd University of Petroleum & Minerals
    Dhahran 31261, Saudi Arabia
    e-mail

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