Embeddings of differential operator rings and Goldie dimension

Abstract

The differential operator ring S = R [ x ; δ ] S = R[x;\delta ] can be embedded in A 1 ( R ) {A_1}(R) , the first Weyl algebra over R R , where R R is a Q {\mathbf {Q}} -algebra and δ \delta is a locally nilpotent derivation on R R . Furthermore A 1 ( R ) {A_1}(R) is again a differential operator ring over the image of S S . We consider similar embeddings of the smash product R # U ( L ) R\# U(L) , where L L is a finite dimensional Lie algebra and U ( L ) U(L) is its universal enveloping algebra. We show that the Weyl algebra over R R has the same Goldie dimension as R R itself and use this to prove that R R and R [ x ; δ ] R[x;\delta ] have the same Goldie dimension where R R is again a Q {\mathbf {Q}} -algebra and δ \delta is locally nilpotent.