Let R be a commutative Noetherian (not necessarily local) ring, I an ideal of R and M a finitely generated R-module. In this paper, by computing the local cohomology modules and Ext-modules via the injective resolution of M, we proved that, if for an integer t > 0, dim$_RH_I^i(M){leq}k$ for ${forall}i$ < t, then $$displaystyleigcup_{i=0}^{j}(Ass_RH_I^i(M))_{{geq}k}=displaystyleigcup_{i=0}^{j}(Ass_RExt_R^i(R/I^n,M))_{{geq}k}$$ for ${forall}j{leq}t$ and ${forall}n$ >0. This shows that${igcup}_{n>0}(Ass_RExt_R^i(R/I^n,M))_{{geq}k}$ is a finite set for ${forall}i{leq}t$. Also, we prove that $displaystyleigcup_{i=1}^{r}(Ass_RM/(x_1^{n_1},x_2^{n_2},{ldots},x_i^{n_i})M)_{{geq}k}=displaystyleigcup_{i=1}^{r}(Ass_RM/(x_1,x_2,{ldots},x_i)M)_{{geq}k}$ if $x_1,x_2,{ldots},x_r$ is M-sequences in dimension > k and $n_1,n_2,{ldots},n_r$ are some positive integers. Here, for a subset T of Spec(R), set $T_{{geq}i}={{p{in}T{mid}dimR/p{geq}i}}$.