Let be an ideal of a commutative Noetherian ring R with identity and let M and N be two finitely generated R-modules. Let t be a positive integer. It is shown that AssRHtMN is contained in the union of the sets AssRExti RMHt−i N , where 0 ≤ i ≤ t. As an immediate consequence, it follows that if either HiN is finitely generated for all i < t or SuppRHiN is finite for all i < t, then AssRHtMN is finite. Also, we prove that if d = pdM and n = dimN are finite, then Hd+n MN is Artinian. In particular, AssRHd+n MN is a finite set consisting of maximal ideals.
