finiteness properties of local cohomology modules for (i, j)-minimax modules

Let r be a commutative noetherian ring and let i and j be two ideals of r. in this paper, we introduce the concept of (i, j)-minimax r-module and it is shown that if m is an (i, j)-minimax r-module and t a non-negative integer such that hi i,j (m) is (i, j)-minimax for all i < t, then for any (i, j)- minimax submodule n of ht i,j (m), the r-module homr(r / i, ht i,j (m) / n) is (i, j)-minimax. as a consequence, it follows that the goldie dimension of ht i,j (m) / n is finite and so the set of associated primes of ht i,j (m) / n is finite. this generalizes the main result of azami, naghipour and vakili [2, theorem 4.2].