Non-Artinian Local Cohomology with Respect to a Pair of Ideals

Let R be a commutative Noetherian ring, I, J two ideals of R, and M an R-module. For a non-negative integer t, we show: (a) If for all j < t and are finite (resp., Artinian), then is finite (resp., Artinian). (b) If for all j < t and are finite (resp., Artinian), then is finite (resp., Artinian). In addition, if (R,𝔪) is a local ring, J a non-nilpotent ideal, and M a finite R-module, then we show that is not Artinian for some i ∈ ℕ₀.