On the Anderson-Badawi $\omega_{R[X]}(I[X])=\omega_R(I)$ conjecture

Abstract

Let $R$ be a commutative ring with an identity different from zero and $n$ be a positive integer. Anderson and Badawi, in their paper on $n$-absorbing ideals, define a proper ideal $I$ of a commutative ring $R$ to be an $n$-absorbing ideal of $R$, if whenever $x_1 \cdots x_{n+1} \in I$ for $x_1, \ldots, x_{n+1} \in R$, then there are $n$ of the $x_i$'s whose product is in $I$ and conjecture that $\omega_{R[X]}(I[X])=\omega_R(I)$ for any ideal $I$ of an arbitrary ring $R$, where $\omega_R(I)= \min \{n\colon\text{$I$ is an $n$-absorbing ideal of $R$}\}$. In the present paper, we use content formula techniques to prove that their conjecture is true, if one of the following conditions hold: The ring $R$ is a Pr\"ufer domain. The ring $R$ is a Gaussian ring such that its additive group is torsion-free. The additive group of the ring $R$ is torsion-free and $I$ is a radical ideal of $R$.