On the cofiniteness of local cohomology modules

Let be a commutative Noetherian ring with identity, be an ideal of and be an -module such that is finitely generated for all . We prove that if for all , then for any and for any submodule of that is either -cofinite or minimax, the -module is -cofinite. This generalizes the main result of Bahmanpour and Naghipour [8, Theorem 2.6]. As a consequence, the Bass numbers and Betti numbers of are finite for all . Also, among other things, we show that if either or , then for each finitely generated -module , the -module is -weakly cofinite, for all and . This generalizes [1, Corollary 2.8].