## On $n$-absorbing rings and ideals

### Volume 147 / 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).