On the Minimaxness and Artinianness Dimensions

Let R be a commutative Noetherian ring, I, J be ideals of R such thatJ ⊆ I, and M be a finitely generated R-module. In this paper, we prove that theinvariants AJI(M) := inf{i ∈ N0 | JtHiI (M) is not Artinian for all t ∈ N0} and inf{i ∈N0 | JtHiI (M) is not minimax for all t ∈ N0} are equal. In particular, we show that theinvariants AII(M) and inf{i ∈ N0 | HiI (M) is not minimax} are equal. We also establishthe local-global principle, AJI(M) = inf{AJRpIRp(Mp)|p ∈ Spec (R)}, in some cases.