On the Anderson-Badawi $\omega _{R[X]}(I[X])=\omega _R(I)$ conjecture

Peyman Nasehpour

Address: Department of Engineering Science, Faculty of Engineering, University of Tehran, Tehran, Iran

E-mail: nasehpour@gmail.com

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 $x_1 \dots x_{n+1} \in I$ 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 $R$.

AMSclassification: primary 13A15; secondary 13B02, 13B25, 13F05.

Keywords: n-absorbing ideals, strongly n-absorbing ideals, polynomial rings, content algebras, Dedekind-Mertens content formula, Prüfer domains, Gaussian algebras, Gaussian rings.

DOI: 10.5817/AM2016-2-71