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 TT and RR have the
same uniform rank. For any right noetherian ring R, it is proved that TT
and RR have the same reduced rank. The type of generalized differential
operator ring considered is any ring extension T ⊇ R generated by a finite
set of elements satisfying a suitable version of the Poincaré-Birkhoff-Witt
Theorem.
