Let
A
A
be the coordinate ring of a smooth affine algebraic variety defined over a field
k
k
. Let
D
D
be the module of
k
k
-linear derivations on
A
A
and form
A
[
D
]
A[D]
, the ring of differential operators on
A
A
, as follows: consider
A
A
and
D
D
as subspaces of
End
k
A
{\operatorname {End}_k}A
(
A
A
acting by left multiplication on itself), and define
A
[
D
]
A[D]
to be the subalgebra generated by
A
A
and
D
D
. Let
rk
D
\operatorname {rk} D
denote the torsion-free rank of
D
D
(that is,
rk
D
=
dim
F
F
⊗
A
D
\operatorname {rk}D = {\dim _F}F{ \otimes _A}D
where
F
F
is the quotient field of
A
A
). The ring
A
[
D
]
A[D]
is a finitely generated
k
k
-algebra so its Gelfand-Kirillov dimension
GK
(
A
[
D
]
)
{\text {GK}}(A[D])
may be defined. The following is proved. Theorem.
GK
(
A
[
D
]
)
=
tr de
g
k
A
+
rk
D
=
2
tr de
g
k
A
{\text {GK}}(A[D]) = {\text {tr de}}{{\text {g}}_k}A + \operatorname {rk} D = 2{\text { tr de}}{{\text {g}}_k}A
. Actually we work in a more general setting than that just described, and although a more general result is obtained, this is the most natural and important application of the main theorem.