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

Let $R$ be a commutative ring with identity and $m$, $n$ be positive integers. We introduce the class of $(m,n)$-prime ideals which lies properly between the classes of prime and $(m,n)$-closed ideals. A proper ideal $I$ of $R$ is called $(m,n)$-prime if for $a,b\in R$, $a^mb\in I$ implies either $a^n\in I$ or $b\in I.$ Several characterizations of this new class with many examples are given. Analogous to primary decomposition, we define the $(m,n)$-decomposition of ideals and show that every ideal in an $n$-Noetherian ring has an $(m,n)$-decomposition. Furthermore, the $(m,n)$-prime avoidance theorem is proved.