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Transactions of the American Mathematical Society

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

 

 

Automorphisms of commutative rings


Author: H. F. Kreimer
Journal: Trans. Amer. Math. Soc. 203 (1975), 77-85
MSC: Primary 13B10
DOI: https://doi.org/10.1090/S0002-9947-1975-0396521-0
MathSciNet review: 0396521
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Abstract: Let $ B$ be a commutative ring with 1, let $ G$ be a finite group of automorphisms of $ B$, and let $ A$ be the subring of $ G$-invariant elements of $ B$. For any separable $ A$-subalgebra $ A'$ of $ B$, the following assertions are proved: (1) $ A'$ is a finitely generated, protective $ A$-module; (2) for each prime ideal $ p$ of $ A$, the rank of $ {A'_p}$ over $ {A_p}$ does not exceed the order of $ G$; (3) there is a finite group $ H$ of automorphisms of $ B$ such that $ A'$ is the subring of $ H$-invariant elements of $ B$. If, in addition, $ A'$ is $ G$-stable, then every automorphism of $ A'$ over $ A$ is the restriction of an automorphism of $ B$, and $ {\operatorname{Hom} _A}(A',A')$ is generated as a left $ A'$-module by those automorphisms of $ A'$ which are the restrictions of elements of $ G$.


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DOI: https://doi.org/10.1090/S0002-9947-1975-0396521-0
Keywords: Automorphism, commutative ring, Galois extension
Article copyright: © Copyright 1975 American Mathematical Society