A new depth and the annihilation of local cohomology modules

Let J subset of I be two proper ideals of a commutative Noetherian ring and M a finitely generated module. Strong relative depth is defined and characterized. It is proved that this depth is just the maximum integer n such that H-I(i)(M) can be annihilated by some power of J for all i <= n. It turns out that the local-global principle for the annihilation of local cohomology modules can be formulated as a natural property of this new depth.