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

Let $R$ be a commutative ring with identity and $m$, $n$ be positive integers. In this paper, we 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 $a,b\in R$, $0\neq a^{m}b\in I$ implies either $a^{n}\in I$ or $b\in I$. We justify several properties and characterizations of weakly $(m,n)$-prime ideals with many supporting examples. Furthermore, we investigate weakly $(m,n)$-prime ideals under various contexts of constructions such as direct products, localizations and homomorphic images. Finally, we discuss the behaviour of this class of ideals in idealization and amalgamated rings.