Automorphisms of commutative rings

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 $G$.