On \(2r\)-ideals in commutative rings with zero-divisors (Q6049727)

Let \(R\) be a commutative ring with identity and \(Z(R)\) its set of zero-divizors. A proper ideal \(I\) of \(R\) is said to be a uniformly \(pr\)-ideal if there exists a positive integer \(n\) such that, whenever \(x,y\in R\) with \(xy\in I\), then \(x^n\in I\) or \(y\in Z(R)\). The order of \(I\) is the smallest positive integer for which the aforementioned property holds. The goal of this paper is to study the uniformly \(pr\)-ideals with order \(\leq 2\), which are called \(2r\)-ideals. After giving several properties and characterizations of such ideals, the authors show that many known classes of ideals are \(2r\)-ideals. They also include the study of \(2r\)-ideals in polynomials rings.