Summary: 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.