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Varieties of Algebras ADVANCES IN MATHEMATICS 8, 163-369 (1972) Varieties of Algebras J. MARSHALL OSBORN Depavtmest of Mathematics, University of Wisconsin, Madison, Wisconsin 53706 Preface ............................ 163 I. The Basic Theory ...................... 166 I. Some Definitions. .................... 166 2. V-Free Algebras and Birkhoff’s Theorem .......... 170 3. Linearization ...................... 174 4. Changing the Domain of Operators ............. 187 II. Identities on Commutative Algebras .............. 201 5. Finding Interesting Identities ............... 201 6. The Peirce Development of an Identity ........... 209 7. Metatheorems on Peirce Development and Examples of Charac- teristic 3 and 5. ..................... 227 8. Associative Bilinear Forms. ................ 244 III. Identities on Noncommutative Algebras. ............ 264 9. Finding Noncommutative Identities. ............ 264 10. Developing Noncommutative Identities ........... 275 11. Associative Algebras Satisfying an Additional Identity ..... 298 IV. Periodic Rings. ....................... 312 12. Classes of Algebras That Do Not Form Varieties ....... 312 13. Basic Facts About Periodic Rings .............. 318 14. Associative Periodic Rings. ................ 324 15. Jordan Periodic Rings .................. 341 16. Periodic Power-Associative Rings and Associative Rings with Involution with Periodic Symmetric Elements ........ 355 PREFACE In this paper, we study varieties of (not necessarily associative) rings and algebras. Although many classes of algebras defined by identities have been studied in the last 25 years, the approach has usually been to study each identity individually. Some steps toward a more unified theory have been taken, particularly in the last five years, but a complete theory does not exist as yet. We attempt in this paper to put together some pieces of a unified theory that exist in the literature as well as many 163 ~0 1972 by Academic Press, Inc. 607/8/2-r 164 OSBORN facts which are known to most people working in the area, but which are not written down anywhere. The development of a general theory of varieties of abstract algebras (P. M. Cohn [I], R. S. Pierce [l]), the success of the application of the concept of varieties to group theory (H. Neumann [l]), and the recent use of a varietal approach by Jacobson in his book on Jordan algebras [3] have motivated us to organize the study of algebras satisfying identities as much as possible around the concept of a variety. It is hoped that the result will serve as a general introduction to the uninitiated as well as another step toward a unified theory for the expert. In choosing what material to include here, we have tried to select those topics that apply to many different varieties of algebras and that seem to us to give the best overall understanding. Thus, for example, we say relatively little about many of those properties which are peculiar to the variety of all associative algebras or the variety of Jordan algebras. In some cases where a topic is lengthy and is treated well elsewhere, we just mention it briefly and give a reference rather than take the space to cover the same ground here. We have tried to include everything which we think should be known by someone who wishes to achieve a good understanding of the area and which is not recorded elsewhere. Inevitably our choice of material reflects some of our prejudices as to what topics are most important and most interesting. In this paper we are a little more interested in discussing and illustra- ting methods than we are in the results themselves. For this reason the material is arranged according to the methods used, and the choice of material was governed equally by the desire to have good illustrations of each method discussed and by the desire to present a body of results which as much as possible both includes the most basic results in the area and is representative of the area as a whole. We hope that the arrangement by method will help the general reader to see what makes the subject tick as well as to provide a place for the specialized reader to begin a study of the specific methods that he may wish to learn to use. Particularly in the case of those methods which are treated at greater length, the reader who is not interested in learning the method in detail may prefer to read just enough to get an idea of the method and then to skim the rest of what we have written on that topic. In cases where we have given a more abstract treatment of a method, the reader with less background may find it helpful either to read one or two special cases of the argument first, or to try in one or two special cases to see what each part of the general argument reduces to. VARIETIES OF ALGEBRAS 165 In Chapter I we cover the principal results characterizing varieties of algebras and connecting related varieties. We give the most basic defini- tions in Section 1, including the definitions of @-algebra, identity, and variety. In Section 2, we prove Birkhoff’s theorem which states that a class V of @-algebras is a variety if and only if it is closed under forming subalgebras, homomorphic images, and complete direct sums. The basic facts of linearization and the proof that a variety of algebras over a field can always be defined by multilinear identities when the field has enough elements can be found in Section 3. We also give there the characteriza- tion of an irreducible identity in terms of its derivatives. The relation between varieties defined by the same identities over different rings of operators is treated in Section 4. Chapters II and III deal with the problems of finding interesting varieties and proving structure theorems for the algebras of a given variety. These questions divide fairly naturally into the commutative case which is considered in Chapter II and the noncommutative case considered in Chapter III. We give a full treatment of the general theory of the Pierce development of an identity on a commutative algebra (Section 6) and th e associated metatheorems (Section 7), since this does not appear anywhere else. The results on associative algebras satisfying additional identities are described in Section I 1. Certain classes of algebras that have been studied-mainly those defined by local properties-do not form varieties. In the first section of Chapter IV we discuss briefly some of these classes and the types of methods that can be used to prove things about them. Since such classes have mainly been considered only in the case of associative algebras, we devote the rest of that chapter to developing the structure of one particu- lar class which is of interest in both associative and nonassociative theory. We call these periodic rings. We define a power-associative ring A to be a periodic ring if for every element a E A there exists an integer n > 1 depending on a such that a’& = a. The theory of periodic rings includes a generalization of the theorem of Albert which states that a finite power- associative division algebra is a field. The first three chapters with the possible exception of Section 1 I are intended to be readable by someone who has had a course in algebra at the graduate level. A little more background in the theory of associative rings would be helpful in Sections 11, 13, 14, and a little background in nonassociative theory would be helpful in Sections 15 and 16, although we shall give references to the main results of associative and nonasso- ciative ring theory that we need. 166 OSBORN Our list of references is not intended to be exhaustive, but rather to provide a place to get started in the relevant literature. For example, instead of referring to the original paper we sometimes refer to a later paper or to a book which contains further information in addition to making reference to the original paper. Rather than try to assign credit for each argument that we give, we shall just list at the end of each sec- tion the references that we used which were not credited in the text as well as those that we think the reader will find most helpful for further study. By checking these references, the reader who is sufficiently inter- ested should have no trouble discovering where any argument or result came from in those cases where we were aware of it coming from another source. We believe that the reader interested in an exhaustive list of references for almost any topic discussed here could compile a fairly comprehensive list of references by starting with the set of references that we give for that topic and closing that set under the two operations of adding the names of all relevant papers mentioned in each reference already in the set, and adding the names of all relevant papers listed in Mathematical Reviews under the names of authors of papers already listed in the set. An extensive list of references can be found in Braun and Koecher [I], Jacobson [3], and Schafer [ 11. I. THE BASIC THEORY 1. Some Dejkitions Although the concepts of an identity and a variety can be defined fairly easily in an informal manner that would be quite sufficient for some parts of this paper, there are some results that we wish to prove that require a more precise and formal definition of these concepts. In order to do this we have to begin with a number of definitions under- lying these two concepts. Let us recall that a groupoid is a set N together with a binary operation defined on N which takes its values in N. We shall denote this operation by juxtaposition, and in the case of an expression that involves more than one product we shall use parentheses to indicate in what order the operations are to be performed.
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