On the Anderson-Badawi \(\omega_{R[X]}(I[X])=\omega_R(I)\) conjecture. (Q2828834)

Let \(R\) be a commutative ring with unity, and let \(n\) be a positive integer. As defined by \textit{D. F. Anderson} and \textit{A. Badawi} [Commun. Algebra 39, No. 5, 1646--1672 (2011; Zbl 1232.13001)], a proper ideal \(I\) of \(R\) is \(n\)-\textit{absorbing} if for every \(n+1\) elements \(x_1,\dots, x_{n+1}\) in \(R\) such that \(\prod_{i=1}^{n+1}x_i\in I\), there exist \(n\) elements among the \(x_i\)'s whose product is in \(I\). The main purpose of this paper is to prove some special cases of the Anderson-Badawi conjecture; \(\omega_{R[X]}(I[X])=\omega_R (I)\) for any proper ideal \(I\) of \(R\), where \(\omega_R (I)=\min\{n:I\) is an \(n\)-absorbing ideal of \(R\}\).NEWLINENEWLINE If \(B\) is a commutative \(R\)-algebra, and \(f\in B\), then, as defined by \textit{J. Ohm} and \textit{D. E. Rush} [Math. Scand. 31, 49--68 (1972; Zbl 0248.13013)], \(c(f)\) (the \textit{content} of \(f\)) is the intersection of the ideals \(I\) of \(R\) such that \(f\in IB\). By using this generalization of the content of a polynomial, the author defines Gaussian and Armendariz algebras as a natural generalization of Gaussian and Armendariz rings. He proves among other results that if \(B\) is a content \(R\)-algebra, then \(B\) is a Gaussian \(R\)-algebra iff \(B/IB\) is an Armendariz \((R/I)\)-algebra for every ideal \(I\) of \(R\). Here is the main result of this paper:NEWLINENEWLINE Let \(B\) be a content \(R\)-algebra, and let \(I\) be an ideal of \(R\). Then \(\omega_B(IB)=\omega_R (I)\) under each of the following three conditions.NEWLINENEWLINE (1) The ring \(R\) is a Prüfer domain.NEWLINENEWLINE (2) The ring \(R\) is a Gaussian ring and its additive group is torsion-free.NEWLINENEWLINE (3) The additive group of the ring \(R\) is torsion-free and \(I\) is a radical ideal of \(R\).