On $n$-absorbing rings and ideals

Abdallah Laradji¹

¹ Department of Mathematics & Statistics King Fahd University of Petroleum & Minerals Dhahran 31261, Saudi Arabia

Colloquium Mathematicum, Tome 147 (2017) no. 2, pp. 265-273.

Résumé

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).

Détail

DOI : 10.4064/cm6844-5-2016

Keywords: proper ideal commutative ring n absorbing resp strongly n absorbing elements resp ideals ldots cdots implies product conjectured anderson badawi n absorbing ideal nbsp strongly n absorbing nbsp n absorbing ideal nbsp mathrm rad subseteq prove these conjectures various classes rings extending several known results n absorbing ideals by product implies nbsp