Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Prime ideals in enveloping rings

Author: D. S. Passman
Journal: Trans. Amer. Math. Soc. 302 (1987), 535-560
MSC: Primary 17B35; Secondary 16A33, 16A66
MathSciNet review: 891634
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Abstract: Let $ L$ be a Lie algebra over the field $ K$ of characteristic 0 and let $ U(L)$ denote its universal enveloping algebra. If $ R$ is a $ K$-algebra and $ L$ acts on $ R$ as derivations, then there is a natural ring generated by $ R$ and $ U(L)$ which is denoted by $ R\char93 U(L)$ and called the smash product of $ R$ by $ U(L)$. The aim of this paper is to describe the prime ideals of this algebra when it is Noetherian. Specifically we show that there exists a twisted enveloping algebra $ U(X)$ on which $ L$ acts and a precisely defined one-to-one correspondence between the primes $ P$ of $ R\char93 U(L)$ with $ P \cap R = 0$ and the $ L$-stable primes of $ U(X)$. Here $ X$ is a Lie algebra over some field $ C \supseteq K$.

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Article copyright: © Copyright 1987 American Mathematical Society