Embeddings of differential operator rings and Goldie dimension
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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.References
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Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 102 (1988), 9-16
- MSC: Primary 16A05,; Secondary 17B30,17B35
- DOI: https://doi.org/10.1090/S0002-9939-1988-0915706-X
- MathSciNet review: 915706