Some results on the annihilators and attached primes of local cohomology modules

Let (R, m) be a local ring and M a finitely generated R-module. It is shown that if M is relative Cohen-Macaulay with respect to an ideal a of R, then Ann(R)(H-a(cd(a, M)) (M)) = Ann(R) M/L = Ann(R) M and Ass(R)(R/Ann(R) M) subset of {p is an element of Ass(R) M vertical bar cd(a, R/p) = cd(a, M)}, where L is the largest submodule of M such that cd(a, L) < cd(a, M). We also show that if H-a(dimM) (M) = 0, then Att(R)(H-a(dimM-1) (M)) = {p is an element of Supp(M)vertical bar cd(a, R/p) = dim M-1}, and so the attached primes of H-a(dimM-1) (M) depend only on Supp(M)vertical bar Finally, we prove that if M is an arbitrary module (not necessarily finitely generated) over a Noetherian ring R with cd(a, M) = cd(a, R/Ann(R)M), then Att(R)(H-a(cd(a, M)) (M)) subset of {p is an element of V(Ann(R) M)| cd(a, R/p) = cd(a, M)}. As a consequence of this, it is shown that if dimM = dim R, then Att(R)(H-a(dimM) (M)) subset of {p is an element of Ass(R) M vertical bar cd(a, R/p) = dim M}.