##
**The concise handbook of algebra.**
*(English)*
Zbl 1008.00004

Dordrecht: Kluwer Academic Publishers. xvi, 618 p. (2002).

This handbook on algebra is intended to make the basic knowledge in various areas of this discipline more accessible. It is introductory in nature (allowing to have the necessary background to deal with more elaborate handbooks like the one of M. Hazewinkel [Handbook of Algebra. Vol 1 (1996; Zbl 0859.00011) and Vol 2 (2000; Zbl 0949.00006)]) with many cross references.

It compromises between a rapid consultation and a more detailed study. Thus it can be seen as a compendium of basic knowledge on algebra. On the other hand, the editors write that they followed their taste in the design of the volume.

The book has an encyclopaedic character but develops the ideas in a streamlined fashion mirroring in this way the above mentioned compromise. One finds nine parts each with its set of sections: Semigroups, groups, rings-modules-algebras, fields, representation theory, lattices, universal algebra, homological algebra, miscellaneous. Each section is written by specific authors. At the end, there are forty pages of bibliography, some information on the authors (addresses, pictures) and important in such a case, a detailed index. Like the parts, the sections start usually with simple situations and become refined afterwards. Typically, from the definitions, one passes to the free case and further to varieties of the corresponding structures, word aspects…The sections end with computational problems and applications. On the way one picks a diversity of attractive topics (Hopf algebras, differential Galois theory, modular representation, algebraic K-theory). The last part contains model theory for algebraists, linear codes, history of algebra.

Basic knowledge, recent developments and applications are combined in a harmonious way. One of the best qualities a book can have is to be useful. This is the case here.

It compromises between a rapid consultation and a more detailed study. Thus it can be seen as a compendium of basic knowledge on algebra. On the other hand, the editors write that they followed their taste in the design of the volume.

The book has an encyclopaedic character but develops the ideas in a streamlined fashion mirroring in this way the above mentioned compromise. One finds nine parts each with its set of sections: Semigroups, groups, rings-modules-algebras, fields, representation theory, lattices, universal algebra, homological algebra, miscellaneous. Each section is written by specific authors. At the end, there are forty pages of bibliography, some information on the authors (addresses, pictures) and important in such a case, a detailed index. Like the parts, the sections start usually with simple situations and become refined afterwards. Typically, from the definitions, one passes to the free case and further to varieties of the corresponding structures, word aspects…The sections end with computational problems and applications. On the way one picks a diversity of attractive topics (Hopf algebras, differential Galois theory, modular representation, algebraic K-theory). The last part contains model theory for algebraists, linear codes, history of algebra.

Basic knowledge, recent developments and applications are combined in a harmonious way. One of the best qualities a book can have is to be useful. This is the case here.

Reviewer: A.Akutowicz (Berlin)

### MSC:

00A20 | Dictionaries and other general reference works |

00A05 | Mathematics in general |

20-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to group theory |

13-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to commutative algebra |

16-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to associative rings and algebras |

12-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to field theory |

06-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to ordered structures |

18-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to category theory |

08-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to general algebraic systems |