FINITE OUTER GALOIS THEORY OF NON-COMMUTATIVE RINGS
Yôichi Miyashita · Hokkaido Mathematical Journal · 1966
\S 1. Galois extension and normal basis.115\S 2. The first characterization of fixed-subrings.118 \S 3. The second characterization of fixed-subrings.121 \S 4. Extension of isomorphisms.122 \S 5. Heredity of Galois extensions.123 \S 6. Completely outer case.126 \S 7. Several results.130 \S 0. Introduction.It is the purpose of this paper to extend the Galois theory of commutative rings given by S. U. Chase, D. K. Harrison and A. Rosenberg [4] to non-commutative case.In what follows, for the sake of simplicity, we shall state main results for directly indecomposable rings: Let $A i 1$ be a directly indecomposable ring, $G$ a finite group of automorphisms of $A$ , and $B=A^{e}=$ { $x\in A;a(x)=x$ for all $\sigma$ in $G.$ }.We call $A/B$ a G-Galois extension if there are elements $a_{1},$ $\cdots,a_{n}$ ; $a_{1}^{*},$ $\cdots,a_{n}^{*}$ in $A$ such that $\sum_{i}a_{t}\cdot\sigma(a_{i}^{*})=$ $\delta_{1,\sigma}(\sigma\in G)$ , where $\delta_{1,\sigma}$ means Kronecker's delta.If $V_{A}(B)=C$ (the center of $A$ ), then $A/B$ is a G-Galois extension if and only if the mapping $x\otimes y\rightarrow xy$ from $A\otimes_{B}A$ to $A$ splits as an A-A-homomorphism (Th.1.5).Let $A/B$ be a G-Galois extension, and $A^{\prime}$ a G-invariant subring of $A,$ $i.e.,$ $\sigma(A^{\prime})=A$ ' for all $\sigma$ in $G$ , and put $B^{\prime}=A^{\prime e}$ .If $A^{\prime}/B^{\prime}$ is a G-Galois extension and $B_{B^{\prime}}^{\prime}$ is a direct summand of $A_{B^{\prime}}^{\prime}$ , then there hold the following.(1) For any subgroup $H$ of $G,$ $A^{H}=B\otimes_{B^{\prime}}A^{\prime H}=A^{\prime H}\otimes_{B^{\prime}}B$ .(2) Let $\{\overline{T}\}$ be the set of all G-invariant inter- mediate rings of $A/A^{\prime}$ , and $\{T\}$ the set of all intermediate rings of $B/B^{\prime}$ such that $A^{\prime}T=TA^{\prime}$ .Then, $\overline{T}\rightarrow\overline{T}\cap B$ and $T\rightarrow A^{\prime}T=TA^{\prime}$ are mutually converse order isomorphisms between $\{\overline{T}\}$ and $\{T\}$ , and $\overline{T}/(\overline{T}\cap B)$ is a G-Galois extension (Th. 5. 1).Let $A/B$ be a G-Galois extension, $V_{A}(B)=C$, and $B_{B}$ a direct summand of $A_{B}$ .Then there hold the following: (1) $G$ coincides with the set of all B-automorphisms of $A$ (Th.4.2).(2) For any subgroup $H$ of $G,$ { $\sigma\in G$ ; $\sigma|A^{H}=1_{A}^{\rho}\}=H$ (3) If $T$ is an intermediate ring of $A/B$, the following areT)$ -projective if and only if there are elements $t_{1},$ $\cdots,$ $t_{n}\in T$ and $a_{1}^{\prime},$ $\cdots,a_{n}^{\prime}\in A^{\prime}$ such that $\sum_{i}t_{t}a_{i}^{\prime}=1$ and $\sum_{i}xt_{i}\otimes a_{i}^{\prime}=\sum_{i}t_{i}\otimes a_{i}^{\prime}x(\in T\otimes_{B^{\prime}}A^{\prime})$ for all $x$ in $T$ .When this is the case, $\{(t_{i}, a_{i}^{\prime})\};i=1,$ $\cdots,$ $n$ } is called a $(B^{\prime}, T)$ -projective coordinate system for $A^{\prime}$ .If $A^{\prime}$ is $(B^{\prime}, A^{\prime})$ -projective, then we call $A^{\prime}/B^{\prime}$ a separable extension.Let $f$ and $g$ be ring homomorphisms from a ring $A^{\prime}$ to a ring $A^{\prime\prime}$ .$f$ and $g$ are called strongly distinct if, for any non-zero central idempotent $e$ of $A^{\prime\prime}$ , there is an element $x$ in $A^{\prime}$ such that $f(x)e eq g(x)e$ .Let $\mathfrak{S}$ be a set of