Book review

The present book is devoted to study the set of automorphisms for some associative rings. This study is very interesting for specialists in rings and modules, in principal because it is a generalization of one of the most beautiful creations of modern algebra, namely the Galois theory. The author, V.K. Kharchenko, has many contributions in this field and his book is the first attempt of collecting main results obtained in the last decades, if we do not take into consideration a good monography published in 1980 in Lecture Notes, by S. Montgomery.

The first two chapters of the book under review contain the groundwork on its topics, namely the Bergman-Isaacs theorem (If a finite group G acts on a semiprime ring R with no additive |G|-torsion, then the fixed ring R^G is also semiprime.), the Martindale quotient ring construction, the structure of primitive rings with nonzero socle (proving that, if R is a prime ring with a generalized polynomial identity, then its central closure is primitive with nonzero socle, such that the endomorphisms ring of a minimal right ideal is finite dimensional over its centroid). The set of the "X-inner" automorphisms of the ring R (X is the first Russian letter of the author's name) is studied in these chapters, as well as the crucial notion of linear dependence for automorphisms and for derivations, by introducing the identities involving them (DA-identities). A system <formula> of DA-identities is essential, if the ideal generated by the values of all <formula> is the whole ring. A nice result given here is the following one: If a ring R with 1 and no additive torsion has an essential system of DA-identities, then R is a PI-ring. For PI-rings, the author describes the Galois extensions. The study goes on in the Chapters 3 and 4, in which the Galois theory of prime rings is built for the case of automorphisms and for the case of derivations. Prime rings with reduced-finite groups of automorphisms are studied in details, obtaining best results for the algebra of the group assumed to be quasi-Frobenius or for the N-groups (N = Noetherian). For a prime ring with an N-group of automorphisms, there exists a bijection between N-subgroups and intermediate subrings satisfying some simple restrictions.

Such a bijection is also found for prime rings of a prime characteristic p and restricted Lie algebras of outer derivations. The generalization goes further to a Galois theory of semiprime rings (Chapter 5), where the most interesting cases are those in which the ring is finite dimensional or the derivations turn out to be inner on the quotient ring.

In the last chapter, the author applies methods developed in the previous sections to problems as fixed elements of a free algebra, finite generation of the algebra of invariants (results from the Koryukin's paper since 1984), the study of the relations of a ring with its fixed ring under a group of automorphisms, the effect of conditions like primitivity, Goldie, indecomposability, Montgomery equivalence, on derivations and on automorphisms.

V.K. Kharchenko has written a book for specialists on rings and algebras; the reader is supposed to know free products, weak algorithm, free relations, generally to have special knowledge on rings and algebras. We have read also the book written in Russian and we consider it one of the best texts in that concerns Galois theory on associative rings. Unfortunately, the translation from Russian to English is totally unsatisfactory; sometimes it is difficult to understand the results because of using wrong words (for example, it is said simple instead of prime, directed instead of inductive and many others). Some mistranslated theorems, due to mixing up definite and indefinite articles, can be hardly restated correctly. The names of non-Russians authors quoted in the book are often wrongly written (in Russian the names are written phonetically!). Even with these unpleasant facts, the book can be useful for specialists.