on the cofiniteness of local cohomology modules

Let r be a commutative noetherian ring with identity, i be an ideal of r and m be an r-module such that extʲ (r/i, m ) is finitely generated for all j. we prove that if dim hⁱ(m ) ≤ 1 for all i, then for any i ≥ 0 and for any submodule n of hⁱ(m ) that is either i-cofinite or minimax, the r-module hⁱ(m )/n is i-cofinite. this generalizes the main result of bahmanpour and naghipour [8, theorem 2.6]. as a consequence, the bass numbers and betti numbers of hⁱ(m ) are finite for all i ≥ 0. also, among other things, we show that if either dim r/i ≤ 2 or dim m ≤ 2, then for each finitely generated r-module n , the r-module extʲ (n, hⁱ(m )) is i-weakly cofinite, for all i ≥ 0 and j ≥ 0. this generalizes [1, r i corollary 2.8].