Let R be a commutative Noetherian ring, I, J be two ideals of R, M be an R-module and S be a Serre class of R-modules.Apositive answer to the Huneke’s conjecture is given for a Noetherian ring R and minimax R-module M of Krull dimension less than 3, with respect to S. There are some results on cofiniteness and Artinianness of local cohomology modules with respect to a pair of ideals. For a ZD-module M of finite Krull dimension and an integer n ∈ N, if Hi I,J (M) ∈ S for all i > n, then Hi I,J (M)/a j Hi I,J (M) ∈ S for any a ∈ ˜W (I, J ), all i ≥ n, and all j ≥ 0. By introducing the concept of Serre cohomological dimension of M with respect to (I, J ), for an integer r ∈ N0, Hj I,J (R) ∈ S for all j > r iff Hj I,J (M) ∈ S for all j > r and any finite R-module M.
