A new generalization of \((m, n)\)-closed ideals (Q6579963)

All the rings considered in this paper are commutative with identity. A well known notion says that for positive integers \(m\) and \(n\), an ideal \(I\) of a ring \(R\) is said \((m,n)\)-closed if \(x\in R\), \(x^m\in I\) implies \(x^n\in I\). In the paper under review, the authors generalize this notion in the following way. A proper ideal \(I\) of \(R\) is called quasi \((m,n)\)-closed if \(x\in R\), \(x^m\in I\) implies \(x^n\in I\) or \(x^{m-n}\in I\). Then they give many properties and characterizations of such ideals illustrated by several examples. Also, they discuss the notion of quasi \((m,n)\)-closed ideals in some constructions like direct product, localization, homomorphic image and idealization.