On weakly \((m, n)\)-prime ideals of commutative rings (Q6589631)

Let \(R\) be a commutative ring with identity and \(m,n\) be positive integers. In this paper, authors introduce the class of weakly \((m,n)\)-prime ideals generalizing \((m,n)\)-prime and weakly \((m,n)\)-closed ideals. A proper ideal \(I\) of \(R\) is called weakly \((m,n)\)-prime if for \(a,b\in R,\) \(0\neq a^mb \in I\) implies either \(a^n\in I\) or \(b\in I.\) Having introduced the notion of weakly \((m,n)\)-prime ideals, authors justify several properties and characterizations of weakly \((m,n)\)-prime ideals with many supporting examples. Furthermore, they investigate weakly \((m,n)\)-prime ideals under various contexts of constructions such as direct products, localizations and homomorphic images. Finally, they discuss the behaviour of this class of ideals in idealization and amalgamated rings.