On Anderson-Badawi conjectures
Let \(R\) be a commutative ring with \(1\neq 0\). As a generalization of prime ideals, a proper ideal \(I\) of \(R\) is called a 2-absorbing ideal of \(R\) if whenever \(a, b, c \in R\) and \(abc \in I\), then \(ab \in I\) or \(ac \in I\) or \(bc \in I\) [\textit{A. Badawi}, Bull. Aust. Math. Soc. 75, No. 3, 417--429 (2007; Zbl 1120.13004)]. More generally, let \(n\) be a positive integer, a proper ideal \(I\) of \(R\) is said to be an \(n\)-absorbing ideal if whenever \(x_1 \cdots x_{n+1} \in I\) for \(x_1,\cdots, x_{n+1} \in R\), then there are \(n\) of the \(x_{i}s\) whose product is in \(I\). See [\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 said to be a strongly \(n\)-absorbing ideal if whenever \(I_1 \cdots I_{n+1} \subseteq I\) for ideals \(I_1, \cdots ,I_{n+1}\) of \(R\), then the product of some \(n\) of the \(I_{i's}\) is in \(I\). In this paper, the authors studied Anderson-Badawi conjectures. These conjectures are: Conjecture 1 states that every \(n\)-absorbing ideal of \(R\) is strongly \(n\)-absorbing and Conjecture 2: \(\mathrm{Rad}(I )^n \subseteq I\), where \(\mathrm{Rad}(I)\) denotes the radical ideal of \(I\). In [\textit{P.-J. Cahen} et al., in: Commutative algebra. Recent advances in commutative rings, integer-valued polynomials, and polynomial functions. Based on mini-courses and a conference on commutative rings, integer-valued polynomials and polynomial functions. New York, NY: Springer. 353--375 (2014; Zbl 1327.13002)], it was conjectured also that \(I[X]\) is an \(n\)-absorbing ideal of the polynomial ring \(R[X]\) for each \(n\)-absorbing ideal of the ring \(R\) (Conjecture 3). The authors gave an answer to (Conjecture 2) for \(n=3,n=4\) and \(n=5\) and they proved that Conjecture 1 and Conjecture 3 hold in various classes of rings.
scientific article; zbMATH DE number 6845416
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