Cohn P.M. Algebra: Volume 3

Second Edition — Chichester: John Wiley & Sons, 1991. — 483 p.Volume 2 of the Second Edition, which included (besides new material) about half of the old Vol. 2, appeared in 1989. With the present volume the work is now complete (although many topics of current interest have had to be omitted). This volume includes the parts of Chs. 1, 3, 4, 6, 7, 8, 10, 12 left over, as well as further new material. Ch. 1 takes over abstract dependence relations and the development of the natural numbers (viewed as algebra with a unary operator) from the old Ch. 1, together with the basic notions on universal algebra. The material on graded algebras from Ch. 3 has been expanded to bring out the role of the universal derivation bimodule and now includes more on Hilbert series (formerly called 'gocha') and forms the new Ch.
2. Chs. 4 (homological algebra) and 6 (further field theory) become 3 and 5 respectively, with minor revisions and some rearrangement. Topics left over from Chs. 7 and 8 (real fields and quadratic forms) now form Ch. 8, where a discussion of the symplectic and orthogonal groups has been added. Ch. 12, on Noetherian and Pi-rings, has become Ch. 9, while Ch. 10 (Artinian rings) now appears in Chs. 6,7 and
10. The final chapter, on skew fields, gives a simplified treatment of the Dieudonne determinant and establishes the existence of 'free fields'. While this topic might be regarded as unduly specialized by some, the proofs have now been so far simplified as to justify their inclusion in a graduate text; in any case the specialization lemma, on which the proof is based, is of independent interest. There is also a discussion of valuations on skew fields and the book ends with an example of a skew field extension of different left and right dimensions (Artin's problem), as an illustration of the preceding theory.Preface to the Second Edition
Conventions on terminology and notes to the reader
Universal algebra
Multilinear algebra
Homological algebra
Further group theory
Further field theory
Algebras
Central simple algebras
Quadratic forms and ordered fields
Noetherian rings and polynomial identities
Rings without finiteness assumptions
Skew fieldsList of notations