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Zbl 0659.16001
Cohn, P.M.
Free rings and their relations. 2nd ed. (English)
[B] London Mathematical Society Monographs, No.19. London etc.: Academic Press (Harcourt Brace Jovanovich, Publishers). XXII, 588 p. {\$} 80.00; \sterling 66.50 (1985).

The second edition of ``Free rings and their relations'' covers not only the material of the first edition (1971), but also the important contributions to the area in the period between the two books. The second edition has 8 chapters, a chapter 0 and two appendices and has almost doubled in size compared to the first edition. \par The first edition has been reviewed in great details in Zbl 0232.16003 and we will mainly comment on the additions, but we will like to mention that the material of the first edition has been rewritten, many improvements in the presentation have been made and a number of new exercises has been added. \par Chapter 0 which mainly consists of background material from ring theory now includes a paragraph on Hermite Rings. \par As major additions to the Chapters 1-4 are ``The Theory of Hilbert Series of a filtered Ring'' and ``Computation of the dependence number'', the dependence number for a ring is an important concept in constructing rings with certain pathological properties. \par Chapter 5: ``Modules over Firs and Semifirs'' has a paragraph on Sylvester domains, the results here are used later in the description of non-full matrices. Also a new version of the ``specialization lemma'' is included in this chapter. \par Chapter 6 has been completely changed from the corresponding chapter in the first edition and substantial extended. It now includes the theory of automorphisms of the ring of polynomials in two indeterminates over a field as well as a proof of the Makar-Limanov Theorem: Automorphisms of $k<x\sb 1,x\sb 2>$ are tame and the natural mapping from $Aut\sb kk<x\sb 1,x\sb 2>$ to $Aut\sb kk[x\sb 1,x\sb 2]$ is an automorphism. In this chapter one also finds recent results of Kharchenko, Dicks and Formanek on the fixed ring of a group of linear automorphisms of $k<x\sb 1,...,x\sb d>$. The chapter finishes by Kharchenko's results on Galois correspondence for X-outer automorphism groups. \par Chapter 7, which is a very central chapter in the book, contains an extensive theory of prime matrix ideals, the universal field of fractions of a semifir and localization in the style of Gerasimov and Malcolmson. \par Chapter 8 now also includes Laurent series and the Malcev-Neumann construction. \par The author has taken great care in presenting the topics of the book and the reviewer finds that the author has succeeded in making a very readable book, which is highly recommended to anyone interested in non commutative ring theory.
[S.Jøndrup]

MSC 2000:: *16-02 Research monographs (assoc. rings and algebras)
16S10 Associative rings determined by universal properties
16W60 Filtrations and valuations, etc. (assoc. rings and algebras)
16P50 Localization and associative Noetherian rings
16S50 Endomorphism rings: matrix rings
16W20 Morphisms of associative rings
16Dxx Modules, bimodules and ideals (assoc. rings and algebras)

Keywords: Hermite Rings; Hilbert Series; dependence number; Firs; Semifirs; Sylvester domains; specialization lemma; automorphisms; Makar-Limanov Theorem; fixed ring; group of linear automorphisms; X-outer automorphism groups; prime matrix ideals; universal field of fractions; Laurent series; Malcev-Neumann construction

Citations: Zbl 0232.16003

Cited in: Zbl 1114.16001 Zbl 1071.16007 Zbl 1170.14310 Zbl 1021.16015 Zbl 0967.16015 Zbl 1021.16019 Zbl 0983.58007 Zbl 0964.57030 Zbl 0902.16037 Zbl 0834.16022 Zbl 0767.16006 Zbl 0751.12003 Zbl 0729.16024 Zbl 0706.16019 Zbl 0676.16013 Zbl 0675.16002 Zbl 0667.16001

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