On the normality of one-fibered monomial ideals (Q2822087)

Let \(R\) be a commutative Noetherian ring with identity 1. Let \(I\) be an ideal of \(R\). \(I\) satisfies the condition \(\left(Z_2\right)\) if there exists an integer \(b\geq 0\) such that for all \(x,y \in R\) and all integers \(n\geq1\), whenever \(xy\in I^{2n+b}\), then either \(x\) or \(y\) belongs to \(I^n\). If \(I\) is a one-fibered ideal of a Noetherian ring, \(b\left(I\right)\) denotes the smallest non-negative integer \(b\) for which the condition \(\left(Z_2\right)\) is satisfied. In this paper the authors presented a generalization to the higher-dimensional situation of the main results of the first author about the normality of one-fibered monomial ideals [\textit{C. Beddani}, Commun. Algebra 41, No. 2, 648--655 (2013; Zbl 1285.13012)]. They showed that if \(I\) is a monomial ideal of \(R= k[x_1,x_2,\cdots ,x_d]\), then \(I\) is normal one-fibered if and only if for all positive integers \(n\) and all \(x,y \in R\) such that \(xy \in I^{2n}\), either \(x\) or \(y\) belongs to \(I^n\).