Abstract. Let k be an algebraically closed field of characteristic zero, and let R = k[x 1 , . . . , xn] 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 D(R/I) on the ring R/I, and I R (I), the idealiser of I in R. We show that D(R/I) and 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 I R (I).To each simplicial complex ∆ on V = {v 1 , . . . , vn} there is a corresponding square-free monomial ideal I ∆ , and the Stanley-Reisner ring associated to ∆ is defined to be k[∆] = R/I ∆ . We find necessary and sufficient conditions on ∆ for D(k[∆]) to be left Noetherian.