UID:
almahu_9949697627202882
Format:
1 online resource (404 p.)
ISBN:
1-281-22272-0
,
9786611222727
,
0-08-052706-X
Series Statement:
North-Holland mathematics studies ; 184
Content:
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 grou
Note:
Description based upon print version of record.
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Cover; Contents; Preface; Introduction; Chapter I. Alternative Rings; 1. Fundamentals; 2. The real quaternions and the Cayley numbers; 3. Generalized quaternion and Cayley-Dickson algebras; 4. Composition algebras; 5. Tensor products; Chapter II. An Introduction to Loop Theory and to Moufang Loops; 1. What is a loop?; 2. Inverse property loops; 3. Moufang loops; 4. Hamiltonian loops; 5. Examples of Moufang loops; Chapter III. Nonassociative Loop Rings; 1. Loop rings; 2. Alternative loop rings; 3. The LC property; 4. The nucleus and centre; 5 . The norm and trace; Chapter IV. RA Loops
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1. Basic properties of RA loops2. RA loops have LC; 3. A description of an RA loop; Chapter V. The Classification of Finite RA Loops; 1. Reduction to indecomposables; 2. Finite indecomposable groups; 3. Finite indecomposable RA loops; 4. Finite RA loops of small order; Chapter VI. The Jacobson and Prime Radicals; 1. Augmentation ideals; 2. Radicals of abelian group rings; 3. Radicals of loop rings; 4. The structure of a semisimple alternative algebra; Chapter VII. Loop Algebras of Finite Indecomposable RA Loops; 1. Primitive idempotents of commutative rational group algebras
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2. Rational loop algebras of finite RA loopsChapter VIII. Units in Integral Loop Rings; 1. Trivial torsion units; 2. Bicyclic and Bass cyclic units; 3. Trivial units; 4. Trivial central units; 5. Free subgroups; Chapter IX. Isomorphisms of Integral Alternative Loop Rings; 1. The isomorphism theorem; 2. Inner automorphisms of alternative algebras; 3. Automorphisms of alternative loop algebras; 4. Some conjectures of H. J. Zassenhaus; Chapter X. Isomorphisms of Commutative Group Algebras; 1. Some results on tensor products of fields; 2. Semisimple abelian group algebras
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3. Modular group algebras of abelian groups4. The equivalence problem; Chapter XI. Isomorphisms of Loop Algebras of Finite RA Loops; 1. Semisimple loop algebras; 2. Rational loop algebras; 3. The equivalence problem; Chapter XII. Loops of Units; 1. Reduction to torsion loops; 2. Group identities; 3. The centre of the unit loop; 4. Describing large subgroups; 5. Examples; Chapter XIII. Idempotents and Finite Conjugacy; 1. Central idempotents; 2. Nilpotent elements; 3. Finite conjugacy; Bibliography; Index; Notation
,
English
Additional Edition:
ISBN 0-444-82438-3
Language:
English
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