ANNIHILATORS OF TOP LOCAL COHOMOLOGY MODULES DEFINED BY A PAIR OF IDEALS

Let R be a commutative Noetherian ring, I, J two proper ideals of R and let M be a non-zero finitely generated R-module with c = cd(I, J, M). In this paper, we first introduce TR(I, J, M) as the largest submodule of M with the property that cd(I, J, TR(I, J, M)) < c and we describe it in terms of the reduced primary decomposition of zero submodule of M. It is shown that AnnR(Hd I,J (M)) = AnnR(M/TR(I, J, M)) and AnnR(Hd I (M)) = AnnR(Hd I,J (M)), whenever R is a local ring, M has dimension d with Hd I,J (M) 6= 0 and J tM ⊆ TR(I, M) for some positive integer t.