Two-sided ideals in the ring of differential operators on a Stanley-Reisner ring

Let R be a Stanley-Reisner ring (that is, a reduced monomial ring) with coefficients in a domain k, and K its associated simplicial complex. Also let D_k(R) be the ring of k-linear differential operators on R. We give two different descriptions of the two-sided ideal structure of D_k(R) as being in bijection with certain well-known subcomplexes of K; one based on explicit computation in the Weyl algebra, valid in any characteristic, and one valid in characteristic p based on the Frobenius splitting of R. A result of Traves [Tra99] on the D_k(R)-module structure of R is also given a new proof and different interpretation using these techniques.