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1st Edition - October 24, 1996

**Editors:** E.G. Goodaire, E. Jespers, C. Polcino Milies

eBook ISBN:

9 7 8 - 0 - 0 8 - 0 5 2 7 0 6 - 2

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… Read more

Immediately download your ebook while waiting for your print delivery. No promo code is needed.

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.

Contents. Preface. Introduction. I. Alternative Rings. Fundamentals. The real quaternions and the

Cayley numbers. Generalized quaternion and Cayley-Dickson algebras. Composition algebras. Tensor

products. II. An Introduction to Loop Theory and to Moufang Loops. What is a loop? Inverse

property loops. Moufang loops. Hamiltonian loops. Examples of Moufang loops. III. Nonassociative

Loop Rings. Loop rings. Alternative loop rings. The LC property. The nucleus and centre. The norm

and trace. IV. RA Loops. Basic properties of RA loops. RA loops have LC. A description of an RA

loop. V. The Classification of Finite RA loops. Reduction to indecomposables. Finite indecomposable

groups. Finite indecomposable RA loops. Finite RA loops of small order. VI. The Jacobson and

Prime Radicals. Augmentation ideals. Radicals of abelian group rings. Radicals of loop rings. The

structure of a semisimple alternative algebra. VII. Loop Algebras of Finite Indecomposable RA

Loops. Primitive idempotents of commutative rational group algebras. Rational loop algebras of finite

RA 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 Loop

Rings. The isomorphism theorem. Inner automorphisms of alternative algebras. Automorphisms of

alternative loop algebras. Some conjectures of H.J. Zassenhaus. X. Isomorphisms of Commutative

Group Algebras. Some results on tensor products of fields. Semisimple abelian group algebras.

Modular group algebras of abelian groups. The equivalence problem. XI. Isomorphisms of Loop

Algebras of Finite RA Loops. Semisimple loop algebras. Rational loop algebras. The equivalence

problem. XII. Loops of Units. Reduction to torsion loops. Group identities. The centre of the unit

loop. Describing large subgroups. Examples. XIII. Idempotents and Finite Conjugacy. Central

idempotents. Nilpotent elements. Finite conjugacy. Bibliography. Index. Notation.

Cayley numbers. Generalized quaternion and Cayley-Dickson algebras. Composition algebras. Tensor

products. II. An Introduction to Loop Theory and to Moufang Loops. What is a loop? Inverse

property loops. Moufang loops. Hamiltonian loops. Examples of Moufang loops. III. Nonassociative

Loop Rings. Loop rings. Alternative loop rings. The LC property. The nucleus and centre. The norm

and trace. IV. RA Loops. Basic properties of RA loops. RA loops have LC. A description of an RA

loop. V. The Classification of Finite RA loops. Reduction to indecomposables. Finite indecomposable

groups. Finite indecomposable RA loops. Finite RA loops of small order. VI. The Jacobson and

Prime Radicals. Augmentation ideals. Radicals of abelian group rings. Radicals of loop rings. The

structure of a semisimple alternative algebra. VII. Loop Algebras of Finite Indecomposable RA

Loops. Primitive idempotents of commutative rational group algebras. Rational loop algebras of finite

RA 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 Loop

Rings. The isomorphism theorem. Inner automorphisms of alternative algebras. Automorphisms of

alternative loop algebras. Some conjectures of H.J. Zassenhaus. X. Isomorphisms of Commutative

Group Algebras. Some results on tensor products of fields. Semisimple abelian group algebras.

Modular group algebras of abelian groups. The equivalence problem. XI. Isomorphisms of Loop

Algebras of Finite RA Loops. Semisimple loop algebras. Rational loop algebras. The equivalence

problem. XII. Loops of Units. Reduction to torsion loops. Group identities. The centre of the unit

loop. Describing large subgroups. Examples. XIII. Idempotents and Finite Conjugacy. Central

idempotents. Nilpotent elements. Finite conjugacy. Bibliography. Index. Notation.

- No. of pages: 386
- Language: English
- Published: October 24, 1996
- Imprint: North Holland
- eBook ISBN: 9780080527062

EJ

Affiliations and expertise

Memorial University of Newfoundland, Department of Mathematics and Statistics, St. John's, Newfoundland, CanadaCP

Affiliations and expertise

Universidade de Sao Paulo, Instituto de Mathemática e Estatística, Sao Paulo, Brazil