Alternative Loop Rings

Edited By

  • E.G. Goodaire
  • E. Jespers, Memorial University of Newfoundland, Department of Mathematics and Statistics, St. John's, Newfoundland, Canada
  • C. Polcino Milies, Universidade de Sao Paulo, Instituto de Mathemática e Estatística, Sao Paulo, Brazil

For the past ten years, alternative loop rings have intrigued mathematicians from a wide cross-section of modern algebra. As a consequence, the theory of alternative loop rings has grown tremendously.

One of the main developments is the complete characterization of loops which have an alternative but not associative, loop ring. Furthermore, there is a very close relationship between the algebraic structures of loop rings and of group rings over 2-groups.

Another major topic of research is the study of the unit loop of the integral loop ring. Here the interaction between loop rings and group rings is of immense interest.

This is the first survey of the theory of alternative loop rings and related issues. Due to the strong interaction between loop rings and certain group rings, many results on group rings have been included, some of which are published for the first time. The authors often provide a new viewpoint and novel, elementary proofs in cases where results are already known.

The authors assume only that the reader is familiar with basic ring-theoretic and group-theoretic concepts. They present a work which is very much self-contained. It is thus a valuable reference to the student as well as the research mathematician. An extensive bibliography of references which are either directly relevant to the text or which offer supplementary material of interest, are also included.

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Book information

  • Published: October 1996
  • Imprint: NORTH-HOLLAND
  • ISBN: 978-0-444-82438-7

Table of Contents

Contents. Preface. Introduction. I. Alternative Rings. Fundamentals. The real quaternions and theCayley numbers. Generalized quaternion and Cayley-Dickson algebras. Composition algebras. Tensorproducts. II. An Introduction to Loop Theory and to Moufang Loops. What is a loop? Inverseproperty loops. Moufang loops. Hamiltonian loops. Examples of Moufang loops. III. NonassociativeLoop Rings. Loop rings. Alternative loop rings. The LC property. The nucleus and centre. The normand trace. IV. RA Loops. Basic properties of RA loops. RA loops have LC. A description of an RAloop. V. The Classification of Finite RA loops. Reduction to indecomposables. Finite indecomposablegroups. Finite indecomposable RA loops. Finite RA loops of small order. VI. The Jacobson andPrime Radicals. Augmentation ideals. Radicals of abelian group rings. Radicals of loop rings. Thestructure of a semisimple alternative algebra. VII. Loop Algebras of Finite Indecomposable RALoops. Primitive idempotents of commutative rational group algebras. Rational loop algebras of finiteRA loops. VIII. Units in Integral Loop Rings. Trivial torsion units. Bicyclic and Bass cyclic units.Trivial units. Trivial central units. Free subgroups. IX. Isomorphisms of Integral Alternative LoopRings. The isomorphism theorem. Inner automorphisms of alternative algebras. Automorphisms ofalternative loop algebras. Some conjectures of H.J. Zassenhaus. X. Isomorphisms of CommutativeGroup Algebras. Some results on tensor products of fields. Semisimple abelian group algebras.Modular group algebras of abelian groups. The equivalence problem. XI. Isomorphisms of LoopAlgebras of Finite RA Loops. Semisimple loop algebras. Rational loop algebras. The equivalenceproblem. XII. Loops of Units. Reduction to torsion loops. Group identities. The centre of the unitloop. Describing large subgroups. Examples. XIII. Idempotents and Finite Conjugacy. Centralidempotents. Nilpotent elements. Finite conjugacy. Bibliography. Index. Notation.