Usually, by Galois theory in class of rings one understands a theorem giving the correspondence between some types of groups of automorphisms and some types of subrings from the class of rings under ...consideration. Starting with the work in the 1970s on the regular action of finite groups on rings, one of the central problem was the study of the properties of rings which are preserved in going from the ring to its subring of invariants and vice versa, from the subring of invariants to the ring itself. The methods developed showed their efficiency also in the investigation of arbitrary automorphisms and derivations, mainly from the point of view of their algebraic independence. In particular, the Galois correspondence was established in the class of semiprime rings for both automorphism groups and Lie algebas of derivations. The development of noncommutative Galois theory was possible as a result of the high level of the structure theory of rings with generalized identities. Although the main difficulties can already be handled for prime rings, it is not always possible to transfer proofs directly to the case of semiprime rings. Very often one has to repeat the same arguments in different situations. This phenomena brought about the proof of a metatheorem stating that many results expressed in the language of mathematical logic can be automatically transferred from the class of prime rings to a large class of semiprime rings. The book is organized as follows. Chapter 1 is introductory. It is devoted to the Baer radical in the structure theory of rings, the Bergman-Isaacs theorem for the nilpotency of rings with a fixed-point-free group action, the Quinn theorem on integrality over fixed subrings, the Martindale theorem for rings with generalized identities and the metatheorem. Chapter 2 deals with the algebraic independence of automorphisms and derivations of semiprime rings. More precisely, it turns out that all algebraic dependences are consequences of the obvious ones, which define the algebraic structure of the sets of automorphisms and derivations. The next three chapters are devoted to Galois theory for automorphisms and derivations, first in the class of prime rings and then, by metatheorem, in the class of semiprime rings. The final chapter contains applications to free associative algebras, the finite generation of the algebra of noncommutative invariants with additional action of symmetric groups, Montgomery equivalence, Hopf algebra actions. A good list of references is given as well.
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Publication and manufacture - Novosibirsk : Naučnaja knjiga, 1996
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