Galois theory of simple rings

Abstract

Introduction. The outer Galois theory, started by Jacobson [8], has been developed rather thoroughly [2; 3; 6; 12; 16]. The general Galois theory, dealing with general groups of automorphisms (with some restrictions though), has been established by Cartan [51 and Jacobson [9] in case of sfields. The purpose of the present paper is to offer a similar theory for simple rings with minimum condition('). The same has been given in fact in Hochschild [7] for simple algebras (finite over their centers). But the method breaks down in the case of general simple rings, infinite over their centers, and a new approach is necessary(2). The writer [14] has recently shown that if A is a simple ring and C is a weakly normal (cf. ?1 below) simple subring of A, then the A-left-, C-right-module A is fully reducible, and he has applied this fact, together with some methods in Dieudonne [6], to obtain a theorem of extension for isomorphisms in simple rings. It turns out that this full reducibility of A, with respect to the left-multiplication of A and the rightmultiplication of C, and some crossed product theorems, proved and used in the older papers by Azumaya and the writer [3; 12; 13; 16], are appropriate means for establishing the Galois theory(') for simple rings. In fact, if A is in particular a sfield, then A is clearly minimal (=irreducible) with respect to any operator domain containing the left-multiplication ring of A, and this fact underlies the Galois theory, as well as many other theories, for sfields. It is replaced, when A is a simple ring, by our A-C-full reducibility. The first section of the present paper gives some preliminary, though fundamental, lemmas on weakly normal simple subrings of a simple ring. In ?2 we introduce regular groups of automorphisms of a simple ring, which are the class of automorphism groups employed in our Galois theory, and consider their invariant systems. Conversely, we consider in ?3 the group of automorphisms leaving a subring, of a certain type, elementwise fixed. The Galois theory follows then in ?4. Although our method is rather different, we follow there the pattern of the algebra case in Hochschild [7 ]. Our theory can