This paper is principally concerned with the question of whether 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 or left noetherian $\bbQ$-algebra, or a right noetherian right fully bounded $\bbQ$-algebra, it is proved that $T_T$ and $R_R$ have the same uniform rank. For any right noetherian ring $R$, it is proved that $T_T$ and $R_R$ have the same reduced rank. The type of generalized differential operator ring considered is any ring extension $T\supseteq R$ generated by a finite set of elements satisfying a suitable version of the Poincarè-Birkhoff-Witt Theorem.
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