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Algebraic structures. (English) Zbl 0831.00002

Braunschweig: Vieweg. ix, 165 p. (1995).
This is a very concentrated text (157 pages), with the ambition to cover the fundamentals of: groups, rings, fields, modules, modern linear algebra, quadratic and alternating forms, ring and field extensions (including Galois theory), Noetherian rings and localization, Dedekind domains and a little number theory, categories, Lie and Clifford algebras, and dimension theory of commutative rings and a bit of logic.
The proofs are very condensed and this is achieved through a careful analysis of the arguments and leaving some details to the exercises. The reader is supposed to have a good background in linear algebra and the author hopes the text will provide a good introduction to modern algebra.
Though the book is well written from the point of view of delivering nude information, it lacks one essential ingredient to be a good one, namely the informal part. Many important results are not used at all further on, so the reader cannot grasp their significance.
The reviewer has the feeling that this text is suitable to a review of the topics after the reader was exposed to the real thing; it could also be helpful in an attempt to organize a course around the topics presented. It may sound like a paradox, but the shortest route into algebra is not always a short one ….

MSC:

00A05 Mathematics in general
13-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra
16-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to associative rings and algebras
18-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to category theory
20-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory
15A27 Commutativity of matrices
15A69 Multilinear algebra, tensor calculus
15A75 Exterior algebra, Grassmann algebras
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