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Keywords:
$n$-absorbing ideals; strongly $n$-absorbing ideals; polynomial rings; content algebras; Dedekind-Mertens content formula; Prüfer domains; Gaussian algebras; Gaussian rings
Summary:
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 $x_1 \dots x_{n+1} \in I$ 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 \lbrace n\colon I \text{is} \text{an} n\text{-absorbing} \text{ideal} \text{of} R\rbrace $. 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 Prüfer 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 $R$.
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