Embeddings of differential operator rings and Goldie dimension

Abstract:

The differential operator ring $S = R[x;\delta ]$ can be embedded in ${A_1}(R)$, the first Weyl algebra over $R$, where $R$ is a ${\mathbf {Q}}$-algebra and $\delta$ is a locally nilpotent derivation on $R$. Furthermore ${A_1}(R)$ 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 $R[x;\delta ]$ have the same Goldie dimension where $R$ is again a ${\mathbf {Q}}$-algebra and $\delta$ is locally nilpotent.