Idealisers and Rings of Differential Operators

It is known that in a right Noetherian ring with finite Krull dimension, the idealiser of a finite intersection of maximal right ideals is right Noetherian with finite Krull dimension. This result is generalised to a finite intersection of right ideals for each of which, essentially, the corresponding factor module is critical and has an endomorphism ring given by a right Ore domain. It is then shown that in the ring of differential operators on a regular affine domain, this result applies to the idealiser of a right ideal generated by the prime ideal of a smooth sub-variety of arbitrary co-dimension, and to the idealiser of a finite intersection of such right ideals, provided the corresponding primes are pairwise co-maximal and of equal height. An application is given to prove that the ring of differential operators on a quotient variety formed by glueing together the images of a smooth subvariety under a finite group action is right Noetherian.