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Transactions of the American Mathematical Society

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

 

 

Trace-like functions on rings with no nilpotent elements


Authors: M. Cohen and Susan Montgomery
Journal: Trans. Amer. Math. Soc. 273 (1982), 131-145
MSC: Primary 16A72; Secondary 16A33, 16A38, 16A74
DOI: https://doi.org/10.1090/S0002-9947-1982-0664033-6
MathSciNet review: 664033
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Abstract: Let $ R$ be a ring with no nilpotent elements, with extended center $ C$, and let $ E$ be the set of idempotents in $ C$. Our first main result is that for any finite group $ G$ acting as automorphisms of $ R $, there exist a finite set $ L \subseteq E$ and an $ {R^G}$-bimodule homomorphism $ \tau :R \to {(RL)^G}$ such that $ \tau (R)$ is an essential ideal of $ {(RE)^G}$. This theorem is applied to show the following: if $ R$ is a Noetherian, affine $ PI$-algebra (with no nilpotent elements) over the commutative Noetherian ring $ A$, and $ G$ is a finite group of $ A$-automorphisms of $ R$ such that $ {R^G}$ is Noetherian, then $ {R^G}$ is affine over $ A$.


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DOI: https://doi.org/10.1090/S0002-9947-1982-0664033-6
Article copyright: © Copyright 1982 American Mathematical Society