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ALTERNATIVE LOOP RINGS NHMS
E.G. Goodaire, E. Jespers, C.Polcino Milies
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. Series: North-Holland Mathematics Studies