Galois theory of prime rings
\textit{V. K. Kharchenko} [Algebra Logika 16, 313-363 (1977; Zbl 0397.16037), English translation Algebra Logic 16, 208-246 (1978)] developed a Galois theory for semiprime rings, specifically, given a semiprime ring R and a group G of automorphisms of R, a correspondence between certain subgroups of G and certain subrings of R containing the ring \(R^ G\) of fixed elements of R under the action of G. The present paper is an exposition and extension of Kharchenko's theory for R a prime ring. The group G is required to be an N-group of automorphisms of R, defined as follows. Let \(G_ 0\) be the subgroup of G consisting of inner automorphisms of R, and B the linear span of all units q in the Martindale ring of quotients Q of R such that conjugation by q is in \(G_ 0\). Then G is an N-group if \([G:G_ 0]<\infty\), B is a semisimple algebra, finite dimensional over the center of Q, and any unit of B defines an inner automorphism in \(G_ 0\). The Galois theory of Kharchenko then determines a correspondence between N-subgroups H of G and intermediate rings S, \(R^ G\subset S\subset R\), which are characterized by four conditions. The authors' development of this theory is somewhat different from Kharchenko's. In the final section of the paper, the authors obtain results relating the structures of R and \(R^ G\) (for example, if R is simple, then \(R^ G\) is a finite direct sum of primitive rings); study when \(R^ G\)- algebra embeddings \(\psi\) :\(S\to R\) of intermediate rings S, \(R^ G\subset S\subset R\), are restrictions of elements of G, relating this condition to the behavior of minimal primes of S; and define a concept of an N-subgroup H being almost normal in G, a condition which characterizes when \(R^ H\) has an N-group of automorphisms with fixed ring \(R^ G\). The paper concludes with a collection of examples which serve to indicate the necessity of various conditions required in the theory.
scientific article; zbMATH DE number 3859353
scientific article