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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Embeddings of differential operator rings and Goldie dimension


Author: Declan Quinn
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
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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\char93 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.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0915706-X
Article copyright: © Copyright 1988 American Mathematical Society