Embeddings of differential operator rings and goldie dimension

The differential operator ring S = R[x; δ] can be embedded in Ai(R), the first Weyl algebra over R, where R is a Q-algebra and δ is a locally nilpotent derivation on R. Furthermore Ai(R) is again a differential operator ring over the image of S. We consider similar embeddings of the smash product R#U(L), where L is a finite dimensional Lie algebra and U(L) is its universal enveloping algebra. We show that the Weyl algebra over R has the same Goldie dimension as R itself and use this to prove that R and R[x; δ] have the same Goldie dimension where R is again a Q-algebra and δ is locally nilpotent.