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Infinite crossed products. (English) Zbl 0662.16001
Pure and Applied Mathematics, 135. Boston, MA etc.: Academic Press., Inc. xii, 468 p. $ 84.50 (1989).
The classical theory of crossed products of a field and its Galois group was introduced by E. Noether in the early thirties. Later on, N. Jacobson introduced the notion of crossed product for skew fields and finite groups. In the sixties A. Bovdi enlarged this construction for arbitrary groups and associative rings. In the past 20 years, the theory of crossed products has enjoyed a period of vigorous developments. These include: Cohen-Montgomery duality, a machine to translate crossed results into the context of group-graded rings; Understanding and computing the symmetric Martindale ring of quotients of prime and semiprime rings; Classifying prime and semiprime crossed products; The Galois theory of prime and semiprime rings; Determining the Grothendieck group of a Noetherian crossed product to settle the zero divisor and Goldie rank conjectures. These topics form the core of this remarkable book and indeed, it will be interesting and useful for all specialists, who study modern algebra.
The book contains nine chapters. Chapter 1 contains many of the basic definitions and proves duality and various versions of Maschke’s theorem. Chapter 2 uses Delta methods, a coset counting technique, to classify the prime and semiprime crossed products. Chapter 3 discusses the left and symmetric Martindale ring of quotients and X-inner automorphisms of rings. Chapters 4 and 5 study prime ideals in crossed products R*G with either G finite or with G polycyclic-by-finite and R right Noetherian. Chapters 6 and 7 are concerned with group actions on rings: existence of fixed points, integrality, prime ideals and Galois theory of prime rings. Finally, Chapters 8 and 9 consider the Grothendieck groups of Noetherian crossed products. In particular this material includes the induction theorem, the zero divisor and Goldie rank conjectures, the Zalesskij and Neroslavskij example and some specific computations.
The book is written in a reasonably self-contained manner and it contains over 200 exercises.
Reviewer: S.V.Mihovski

16-02 Research exposition (monographs, survey articles) pertaining to associative rings and algebras
16S34 Group rings
16W20 Automorphisms and endomorphisms
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16P40 Noetherian rings and modules (associative rings and algebras)
16W50 Graded rings and modules (associative rings and algebras)