Let R be a Noetherian ring, and I and J be two ideals of R. Let S be a Serre subcategory of the category of R-modules satisfying the condition CI and M be a ZD-module. As a generalization of the S-depth(I,M) and depth(I, J,M), the S-depth of (I, J) on M is defined as S-depth(I, J,M) = inf{S-depth(a,M) : a 2 fW(I, J)}, and some properties of this concept are investigated. The relations between S-depth(I, J,M) and Hi I,J(M) are studied, and it is proved that S-depth(I, J,M) = inf{i : Hi I,J(M) /2 S}, where S is a Serre subcategory closed under taking injective hulls. Some conditions are provided that local cohomology modules with respect to a pair of ideals coincide with ordinary local cohomology modules under these conditions. Let SuppRHi I,J (M) be a finite subset of Max(R) for all i < t, where M is an arbitrary R-module and t is an integer. It is shown that there are distinct maximal ideals m1,m2, . . . ,mk 2 W(I, J) such that Hi I,J (M) = Hi m1 (M) Hi m2 (M) . . . Hi mk (M) for all i < t.
