##
**Further algebra and applications.
Revised ed. of Algebra, 2nd ed. (1982–1991).**
*(English)*
Zbl 1006.00001

London: Springer. xii, 451 p. EUR 79.95/net; sFr. 133.00; £45.00; $ 79.95 (2003).

The book under review is the second volume of a new and revised edition of P. M. Cohn’s classic three-volume text “Algebra”. The three volumes of the original edition were first published successively in the 1970’s (see Zbl 0272.00003 and Zbl 0341.00002) and, due to their excellence and popularity, have been widely regarded as being among the most outstanding introductory algebra textbooks ever since. Their second, then already reworked edition appeared around 1990. Now, more than ten years later, P. M. Cohn’s classic algebra text became out of print, again, and the author took this fact as an incentive to bring out a new version of it, this time in only two volumes and in a both reworked and updated fashion.

This new second volume, entitled “Further Algebra and Applications”, follows on the subject matter treated in the new first volume “Basic Algebra” (Springer London, 2002; Zbl 1003.00001). Basically, these two new books represent the contents of the former volumes 2 and 3 of the author’s standard text. However, the contents have been rearranged a little, with most of the more advanced topics and applications in the present volume, while the basic theories are pursued further in “Basic Algebra”, just as the two new titles portend. In general, the text has been revised, some exercises have been added and, of course, blemishes have been correlated. The text of this second volume assumes an acquaintance with much of the material covered in “Basic Algebra”, to which reference is made throughout the entire book under review. However, this does not mean that the present volume had forfeit its largely self-contained character. On the contrary, any reader with a profound basic knowledge of modern algebra will find this more advanced and specialized text rather independent of other books and nearly self-contained.

As for the contents of the present volume, there are again eleven chapters, like in “Basic Algebra”, which cover the following topics:

Chapter 1 introduces to the fundamental concepts of universal algebra, including general \(\Omega\)-algebras, congruences and the general isomorphism theorems, free algebras and varieties of \(\Omega\)-algebras, Newman’s diamond lemma, ultraproducts, and an axiomatic development of the natural numbers from the viewpoint of algebraic systems.

Chapter 2 presents some more material from homological algebra. The definition of abelian categories and of functors between them is followed by an abstract description of module categories, including the notion of homological dimension. Derived functors are discussed and exemplified by the construction of the functors Ext and Tor for modules. Finally, universal derivations are explained and then used to prove Hilbert’s syzygy theorems.

Chapter 3, entitled “Further Group Theory”, deals with some advanced topics of general interest and importance in this context. This nice selection of topics, chosen with a particular view to later applications, includes ideas from the theory of group extensions, an account of Hall subgroups, Burnside’s theorem on finite groups, and a discussion of free groups. This is then exemplified by illustrating special classes of groups such as linear groups, symplectic groups, and orthogonal groups.

Chanter 4 continues the study of algebras begun in “Basic Algebra” and covers finitely generated modules over Artinian rings (with the Krull-Remak-Schmidt theorem), projective covers, semiperfect rings, equivalences of module categories, Morita theory, projective, injective and flat modules, separable algebras and, at the end, the Hochschild cohomology of algebras.

Chapter 5 turns to the theory of central simple algebras. Matrix rings over division algebras, simple Artinian rings, quaternion algebras, the Brauer group, crossed products, base change properties, and cyclic algebras are the main topics concisely treated here.

In Chapter 6 the theory of algebras is used to give an account of group representations. While the first four sections of this chapter provide an introduction to the representation theory of finite groups based on the Wedderburn theorems, followed by a special section on complex representations, the rest of the chapter deals with the representation theory of permutation grougs, induced representations, and with the classical theorems of Burnside and Frobenius on finite groups.

Noetherian rings and polynomial identities form the theme of Chapter 7. The author presents some of the less common highlights such as noncommutative princidal ideal domains, skew polynomial rings and Laurent series, A. Goldie’s reduction theorem, PI-algebras (polynomial identity algebras) and their varieties, generic matrix rings and central polynomials, Latyshev’s beautiful proof of Regev’s theorem on tensor products of PI-algebras, and the generalized polynomial identity theorem, together with an application to Amitsur’s theorem on skew fields.

Chapter 8 comes with the title “Rings without Finiteness Assumption” and is devoted to primitive rings, semiprimitive rings, non-unital algebras, semiprime rings, prime PI-Algebras, and free ideal rings. This includes a proof of the general density theorem, which serves as a basic tool in this chapter, and the various radicals in those more general rings.

Chapter 9, on skew fields, gives a simplified treatment of the Dieudonné determinant over a skew field, followed by a much simplified proof of the existence of free fields. The author also examines the concepts of valuation theory over skew fields and touches upon E. Artin’s problem on skew field extensions. As for the latter, the study of pseudo-linear extensions is the essential ingredient used here.

The final two chapters of the book present applications of a different kind.

Chapter 10 discusses some aspects of coding theory. Starting with a sketch of the background material from the theory of error-correcting codes, leading up to the statement of C. Shannon’s classical theorem (1948), the author explains consecutively block codes, linear codes, cyclic codes, and various other codes.

Chapter 11 concludes the book with algebraic language theory and the related topics of variable-length codes, abstract automata and power series rings. More precisely, the author guides the reader into the basics on monoids and monoid actions, languages and grammars, logical machines (automata), variable-length codes (à la McMillan, Schützenberger, and others), and their relations to the theory of free algebras and formal power series rings.

Although the last two chapters can only take the first steps into the respective subject, they nevertheless go far enough to demonstrate how techniques from coding theory are successfully used in the study of free algebras.

As in the first volume, “Basic Algebra”, the author has spent much effort at including a plentiful supply of worked concrete examples and carefully selected exercises.

Apart from the author’s well-known methodical and didactic mastery as a textbook writer, it is especially the wealth of specific, non-standard topics and important applications that makes this algebra text, and in particular this second volume of it, highly unique and valuable. Concentrating on the respective essentials of each topic touched upon, in a just as concise as lucid manner, the author manages to cover a vast spectrum of concepts, methods, principles, aspects, and applications of modern algebra in a masterly style.

P. M. Cohn knows perfectly well how to inspire, fascinate and universally educate the reader. This introductory algebra text remains one of the very best available, all the more so after the profitable revision that it has undergone in this new edition.

This new second volume, entitled “Further Algebra and Applications”, follows on the subject matter treated in the new first volume “Basic Algebra” (Springer London, 2002; Zbl 1003.00001). Basically, these two new books represent the contents of the former volumes 2 and 3 of the author’s standard text. However, the contents have been rearranged a little, with most of the more advanced topics and applications in the present volume, while the basic theories are pursued further in “Basic Algebra”, just as the two new titles portend. In general, the text has been revised, some exercises have been added and, of course, blemishes have been correlated. The text of this second volume assumes an acquaintance with much of the material covered in “Basic Algebra”, to which reference is made throughout the entire book under review. However, this does not mean that the present volume had forfeit its largely self-contained character. On the contrary, any reader with a profound basic knowledge of modern algebra will find this more advanced and specialized text rather independent of other books and nearly self-contained.

As for the contents of the present volume, there are again eleven chapters, like in “Basic Algebra”, which cover the following topics:

Chapter 1 introduces to the fundamental concepts of universal algebra, including general \(\Omega\)-algebras, congruences and the general isomorphism theorems, free algebras and varieties of \(\Omega\)-algebras, Newman’s diamond lemma, ultraproducts, and an axiomatic development of the natural numbers from the viewpoint of algebraic systems.

Chapter 2 presents some more material from homological algebra. The definition of abelian categories and of functors between them is followed by an abstract description of module categories, including the notion of homological dimension. Derived functors are discussed and exemplified by the construction of the functors Ext and Tor for modules. Finally, universal derivations are explained and then used to prove Hilbert’s syzygy theorems.

Chapter 3, entitled “Further Group Theory”, deals with some advanced topics of general interest and importance in this context. This nice selection of topics, chosen with a particular view to later applications, includes ideas from the theory of group extensions, an account of Hall subgroups, Burnside’s theorem on finite groups, and a discussion of free groups. This is then exemplified by illustrating special classes of groups such as linear groups, symplectic groups, and orthogonal groups.

Chanter 4 continues the study of algebras begun in “Basic Algebra” and covers finitely generated modules over Artinian rings (with the Krull-Remak-Schmidt theorem), projective covers, semiperfect rings, equivalences of module categories, Morita theory, projective, injective and flat modules, separable algebras and, at the end, the Hochschild cohomology of algebras.

Chapter 5 turns to the theory of central simple algebras. Matrix rings over division algebras, simple Artinian rings, quaternion algebras, the Brauer group, crossed products, base change properties, and cyclic algebras are the main topics concisely treated here.

In Chapter 6 the theory of algebras is used to give an account of group representations. While the first four sections of this chapter provide an introduction to the representation theory of finite groups based on the Wedderburn theorems, followed by a special section on complex representations, the rest of the chapter deals with the representation theory of permutation grougs, induced representations, and with the classical theorems of Burnside and Frobenius on finite groups.

Noetherian rings and polynomial identities form the theme of Chapter 7. The author presents some of the less common highlights such as noncommutative princidal ideal domains, skew polynomial rings and Laurent series, A. Goldie’s reduction theorem, PI-algebras (polynomial identity algebras) and their varieties, generic matrix rings and central polynomials, Latyshev’s beautiful proof of Regev’s theorem on tensor products of PI-algebras, and the generalized polynomial identity theorem, together with an application to Amitsur’s theorem on skew fields.

Chapter 8 comes with the title “Rings without Finiteness Assumption” and is devoted to primitive rings, semiprimitive rings, non-unital algebras, semiprime rings, prime PI-Algebras, and free ideal rings. This includes a proof of the general density theorem, which serves as a basic tool in this chapter, and the various radicals in those more general rings.

Chapter 9, on skew fields, gives a simplified treatment of the Dieudonné determinant over a skew field, followed by a much simplified proof of the existence of free fields. The author also examines the concepts of valuation theory over skew fields and touches upon E. Artin’s problem on skew field extensions. As for the latter, the study of pseudo-linear extensions is the essential ingredient used here.

The final two chapters of the book present applications of a different kind.

Chapter 10 discusses some aspects of coding theory. Starting with a sketch of the background material from the theory of error-correcting codes, leading up to the statement of C. Shannon’s classical theorem (1948), the author explains consecutively block codes, linear codes, cyclic codes, and various other codes.

Chapter 11 concludes the book with algebraic language theory and the related topics of variable-length codes, abstract automata and power series rings. More precisely, the author guides the reader into the basics on monoids and monoid actions, languages and grammars, logical machines (automata), variable-length codes (à la McMillan, Schützenberger, and others), and their relations to the theory of free algebras and formal power series rings.

Although the last two chapters can only take the first steps into the respective subject, they nevertheless go far enough to demonstrate how techniques from coding theory are successfully used in the study of free algebras.

As in the first volume, “Basic Algebra”, the author has spent much effort at including a plentiful supply of worked concrete examples and carefully selected exercises.

Apart from the author’s well-known methodical and didactic mastery as a textbook writer, it is especially the wealth of specific, non-standard topics and important applications that makes this algebra text, and in particular this second volume of it, highly unique and valuable. Concentrating on the respective essentials of each topic touched upon, in a just as concise as lucid manner, the author manages to cover a vast spectrum of concepts, methods, principles, aspects, and applications of modern algebra in a masterly style.

P. M. Cohn knows perfectly well how to inspire, fascinate and universally educate the reader. This introductory algebra text remains one of the very best available, all the more so after the profitable revision that it has undergone in this new edition.

Reviewer: Werner Kleinert (Berlin)

### MSC:

00A05 | Mathematics in general |

06-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordered structures |

12-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory |

18-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to category theory |

20-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory |

16-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to associative rings and algebras |

68Q45 | Formal languages and automata |

94Bxx | Theory of error-correcting codes and error-detecting codes |