Associated primes of local cohomology modules of weakly Laskerian modules

The notion of weakly Laskerian modules was introduced recently by the authors. Let R be a commutative Noetherian ring with identity, alpha an ideal of R, and M a weakly Laskerian module. It is shown that if alpha is principal, then the set of associated primes of the local cohomology module H-alpha(i) (M)is finite for all i >= 0. We also prove that when R is local, then Ass(R) (H-alpha(i)(M)) is finite for all i >= 0 in the following cases: (1) dim R <= 3, (2) dim R/alpha <= 1, (3) M is Cohen-Macaulay, and for any ideal b with l = grade (b,M), HomR (R/b, H-b(l+l) (M)) is weakly Laskerian.