Attached primes and annihilators of top local cohomology modules defined by a pair of ideals

Assume that R is a complete Noetherian local ring and M is a non-zero finitely generated R-module of dimension n = dim(M) >= 1. It is shown that any non-empty subset T of Assh(M) can be expressed as the set of attached primes of the top local cohomology modules H-I,J(n)(M) for some proper ideals I, J of R. Moreover, for ideals I, J = boolean AND(p is an element of AttR(HIn (M))) p and J' of R it is proved that T = Att(R)(H-I,J(n) (M)) = Att(R)(H-I,J'(n) (M)) if and only if J' subset of J. Let H-I,J(n) (M) not equal 0. It is shown that there exists Q is an element of Supp(M) such that dim(R/Q) = 1 and H-Q(n) (R/p) 6= 0, for each p is an element of Att(R)(H-I,J(n) (M)). In addition, we prove that if I and J are two proper ideals of a Noetherian local ring R, then Ann(R)(H-I,J(n) (M)) = Ann(R)(M/T-R(I, J, M)), where T-R(I, J, M) is the largest submodule of M with cd(I, J, T-R(I, J, M)) < cd(I, J, M), here cd(I, J, M) is the cohomological dimension of M with respect to I and J. This result is a generalization of [1, Theorem 2.3] and [2, Theorem 2.6].