##
**Alternative loop rings.**
*(English)*
Zbl 0878.17029

North-Holland Mathematics Studies. 184. Amsterdam: North-Holland. xv, 387 p. (1996).

Since the beginning of this century group algebras have been of permanent interest. In a similar way as for groups, quasigroups and loop algebras may be defined. However, this construction is too general to produce something remarkable. But the restriction to loop algebras satisfying the left and right alternative laws created a new fertile subject. This book is now the first survey of the theory of alternative loop rings written by the protagonists of this new area.

In the first three chapters the authors present important results on the fundamental algebraic structures the book is concerned with, namely alternative rings, Moufang loops, and loop rings.

Chapter IV investigates the properties of those (finite) loops which, in characteristic different from two, have alternative, but not associative loop rings. Such loops are called RA (ring alternative) loops. Chapter V establishes the classification of finite RA loops.

The following chapters contain an analysis of the loop rings themselves and of the group rings of certain groups which are closely related to RA loops. In Chapter VI, the authors describe the Jacobson and prime radicals of an alternative loop ring. Moreover, it turns out that over many fields, alternative loop algebras are direct sums of simple algebras (i.e., a Maschke type theorem holds). Chapter VII describes the simple components of a (semisimple) alternative loop algebra over the rationals by establishing concrete isomorphisms with Zorn’s vector matrix algebras. In this chapter the authors also treat the primitive idempotents of the group algebra of a finite abelian group. This enables them to characterize the primitive central idempotents of the rational loop algebra \(QL\) of a finite RA loop \(L\) and thus the loop algebra \(QL\) itself concretely.

With Chapter VIII a study of the units of an integral alternative loop ring starts. The authors investigate under what conditions all the units and all the torsion units are trivial. This yields new proofs of well known theorems of G. Higman and S. D. Berman for group rings.

Chapters IX to XI are concerned with isomorphism problems. In Chapter IX it is shown that any finite RA loop is determined by its integral loop ring. Moreover, for a finite RA loop \(L\), every normalized automorphism of \(ZL\) is the composition of an automorphism of \(L\) and an inner automorphism of the rational loop algebra \(QL\). Also variations for loop rings of three conjectures of H. Zassenhaus for group rings are presented.

The entire Chapter X is devoted to the isomorphism problem for group algebras of finite abelian groups since abelian groups play an important role in the structure of RA loops. These results are applied in Chapter XI to loop algebras of RA loops over arbitrary fields.

In Chapter XII the authors collect some more results on the units of an integral alternative loop ring. The problem of finding all the units of an integral alternative loop ring is very hard. Thus the authors confine themselves to more accessible questions, e.g., they exhibit a certain subloop of finite index in the unit loop of \(ZL\).

The theme of the last Chapter XIII is the question under what conditions each element of an alternative loop algebra over a field has only finitely many conjugates. In solving this problem, the authors give conditions under which all idempotents of a loop algebra are central and determine when all the nilpotent elements are trivial.

This book is a competent source for all themes concerning RA loops and their loop rings. It will certainly become a standard reference for specialists working in this area. But the topics of this book may also be interesting for other disciplines. So group and ring theorists will find some new insights and some new proofs of well known theorems for group rings. Since the text is nearly self-contained, the book serves also as an excellent introduction to the theory of alternative loop rings.

In the first three chapters the authors present important results on the fundamental algebraic structures the book is concerned with, namely alternative rings, Moufang loops, and loop rings.

Chapter IV investigates the properties of those (finite) loops which, in characteristic different from two, have alternative, but not associative loop rings. Such loops are called RA (ring alternative) loops. Chapter V establishes the classification of finite RA loops.

The following chapters contain an analysis of the loop rings themselves and of the group rings of certain groups which are closely related to RA loops. In Chapter VI, the authors describe the Jacobson and prime radicals of an alternative loop ring. Moreover, it turns out that over many fields, alternative loop algebras are direct sums of simple algebras (i.e., a Maschke type theorem holds). Chapter VII describes the simple components of a (semisimple) alternative loop algebra over the rationals by establishing concrete isomorphisms with Zorn’s vector matrix algebras. In this chapter the authors also treat the primitive idempotents of the group algebra of a finite abelian group. This enables them to characterize the primitive central idempotents of the rational loop algebra \(QL\) of a finite RA loop \(L\) and thus the loop algebra \(QL\) itself concretely.

With Chapter VIII a study of the units of an integral alternative loop ring starts. The authors investigate under what conditions all the units and all the torsion units are trivial. This yields new proofs of well known theorems of G. Higman and S. D. Berman for group rings.

Chapters IX to XI are concerned with isomorphism problems. In Chapter IX it is shown that any finite RA loop is determined by its integral loop ring. Moreover, for a finite RA loop \(L\), every normalized automorphism of \(ZL\) is the composition of an automorphism of \(L\) and an inner automorphism of the rational loop algebra \(QL\). Also variations for loop rings of three conjectures of H. Zassenhaus for group rings are presented.

The entire Chapter X is devoted to the isomorphism problem for group algebras of finite abelian groups since abelian groups play an important role in the structure of RA loops. These results are applied in Chapter XI to loop algebras of RA loops over arbitrary fields.

In Chapter XII the authors collect some more results on the units of an integral alternative loop ring. The problem of finding all the units of an integral alternative loop ring is very hard. Thus the authors confine themselves to more accessible questions, e.g., they exhibit a certain subloop of finite index in the unit loop of \(ZL\).

The theme of the last Chapter XIII is the question under what conditions each element of an alternative loop algebra over a field has only finitely many conjugates. In solving this problem, the authors give conditions under which all idempotents of a loop algebra are central and determine when all the nilpotent elements are trivial.

This book is a competent source for all themes concerning RA loops and their loop rings. It will certainly become a standard reference for specialists working in this area. But the topics of this book may also be interesting for other disciplines. So group and ring theorists will find some new insights and some new proofs of well known theorems for group rings. Since the text is nearly self-contained, the book serves also as an excellent introduction to the theory of alternative loop rings.

Reviewer: Huberta Lausch (Würzburg)

### MSC:

17D05 | Alternative rings |

20N05 | Loops, quasigroups |

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

16S36 | Ordinary and skew polynomial rings and semigroup rings |

16U60 | Units, groups of units (associative rings and algebras) |