zbMATH — the first resource for mathematics

Commutator theory for congruence modular varieties. (English) Zbl 0636.08001
London Mathematical Society Lecture Note Series, 125. Cambridge etc.: Cambridge University Press. III, 227 p. (1987).
Main contents: Introduction to the commutator in groups and rings. The commutator of congruences in modular varieties of universal algebras. Abelian algebras. Abelian varieties and the associated rings. Structure and representation in modular varieties. A finite basis theorem (that a finite nilpotent algebra of finite type which is a direct product of algebras of prime power order has a finite basis for its identities - this generalizes a result of M. R. Vaughan-Lee [Lect. Notes Math. 1004, 293-308 (1983; Zbl 0525.08008)]).
This book is suitable for a graduate quarter or semester in general algebra; there are exercises, a section on “related literature”, an index, and a list of quoted literature. A manuscript on the same subject and by the same authors has widely circulated and was often quoted in papers by several people, since 1980. Its title: “The commutator and its identities”. Beside its destination as a textbook, these lecture notes are welcome to researchers in abstract and universal algebra as a reference handbook showing the state of the art in commutator theory up to 1986, as far as the foundations of the theory are concerned. A number of very important and far-reaching applications are quoted in the related literature section. We will not repeat here the definition of the commutator operation on congruences, because it would need too much space.
The book is plainly written, from a stylistic point of view. What does not imply that it is plainly readable, because the subject is not among the easiest in modern algebra. Though a young student would find it helpful to get further explanations by a teacher, s/he will find it more profitable to work with it for him/herself. In doing so s/he will neither find any substantial mistake nor any important misprint. One thing was not so clear to the reviewer: the import of exercise No.10, page 56.
Reviewer: A.Ursini

08-02 Research exposition (monographs, survey articles) pertaining to general algebraic systems
08B10 Congruence modularity, congruence distributivity
08A30 Subalgebras, congruence relations