Differential operators on Stanley-Reisner rings

Let $k$ be an algebraically closed field of characteristic zero, and let $R=k[x_{1},\dots ,x_{n}]$ be a polynomial ring. Suppose that $I$ is an ideal in $R$ that may be generated by monomials. We investigate the ring of differential operators $\mathcal {D}(R/I)$ on the ring $R/I$, and $\mathcal {I}_{R}(I)$, the idealiser of $I$ in $R$. We show that $\mathcal {D}(R/I)$ and $\mathcal {I}_{R}(I)$ are always right Noetherian rings. If $I$ is a square-free monomial ideal then we also identify all the two-sided ideals of $\mathcal {I}_{R}(I)$. To each simplicial complex $\Delta$ on $V=\{v_{1},\dots ,v_{n}\}$ there is a corresponding square-free monomial ideal $I_{\Delta }$, and the Stanley-Reisner ring associated to $\Delta$ is defined to be $k[\Delta ]=R/I_{\Delta }$. We find necessary and sufficient conditions on $\Delta$ for $\mathcal {D}(k[\Delta ])$ to be left Noetherian.