Uniform rank over differential operator rings and Poincaré-Birkhoff-Witt extensions

This paper is principally concerned with the question of when a generalized differential operator ring T over a ring R must have the same uniform rank (Goldie dimension) or reduced rank as R, and with the corresponding questions for induced modules. In particular, when R is either a right and left noetherian Q-algebra, or a right noetherian right fully bounded Q-algebra, it is proved that Tτ and RR have the same uniform rank. For any right noetherian ring R, it is proved that Tτ and RR have the same reduced rank. The type of generalized differential operator ring considered is any ring extension T D R generated by a finite set of elements satisfying a suitable version of the PoincareBirkhoff-Witt Theorem.