(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 ? R, amb ? I implies either an ? I or b ? 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.