Uniform Rank over Differential Operator Rings and Poincarè-Birkhoff-Witt Extensions

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Abstract (in LaTex)

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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