This text has been designed for two courses. The first of these is a detailed exposition of the theory of alternative algebras, which except for the finite-dimensional case in [R. D. Schafer, An introduction to nonassociative algebras. New York etc.: Academic Press (1966; Zbl 0145.25601)] could until now only be found in journals. The second and shorter course is a presentation of Jordan algebras, with emphasis on results obtained during the past decade.
The first chapter of this book contains reference material consisting of the basic facts on varieties of algebras and linearization of identities.
Chapter 2 is then devoted to the study of composition algebras. In addition to establishing numerous properties for such algebras, the Cayley-Dickson process is described, followed by a proof of the generalized Hurwitz theorem. With some characteristic 2 exceptions, it is also proved that every simple quadratic alternative algebra is a composition algebra.
Jordan algebras are introduced in Chapters 3 and 4. In addition to basic definitions and examples, the first of these chapters contains proofs of several well-known results concerning special and exceptional Jordan algebras. These results, due to Albert, Cohn, Macdonald, and Shirshov, are the same as those found in Chapter 1 of [N. Jacobson, Structure and representations of Jordan algebras. Providence, R.I.: AMS (1968; Zbl 0218.17010)].
In Chapter 4 the authors study the relationship between solvability and nilpotency in Jordan algebras. The chapter begins with examples of solvable algebras of index 2 that are not nilpotent, even though they satisfy the identity \(x^3=0\). This is followed by a proof of Zhevlakov’s theorem that every solvable finitely-generated Jordan algebra is nilpotent. Among other results included in this chapter is a recent one due to Skosyrskiĭ. Namely, if \(J\) is a special Jordan algebra and \(A\) is an associative enveloping algebra for \(J\), then the locally nilpotent radicals \(\mathcal L(J)\) and \(\mathcal L(A)\) of \(J\) and \(A\) satisfy the relation \(\mathcal L(J) = J \cap\mathcal L(A)\). This means that if \(J\) is special, then so is \(J/\mathcal L(J)\).
In Chapter 5 is presented Shirshov’s combinatoric approach to the well-known Kurosh problem: “If \(A\) is an algebra over an associative-commutative ring \(\Phi\) with \(1\) such that each element of \(A\) is algebraic over some fixed ideal \(Z\) of \(\Phi\), must \(A\) be locally finite over \(Z\)?” For special Jordan or alternative PI-algebras, it is shown the answer to this question is yes. In particular, it thus follows that special Jordan or alternative nil-algebras of bounded index are locally nilpotent.
The next topic considered is solvability and nilpotency in alternative algebras. First there is the Nagata-Higman theorem for associative algebras. This is followed by a modified version of Dorofeev’s example of a solvable but not nilpotent alternative algebra. Then the authors prove Zhevlakov’s generalization of the Nagata-Higman theorem, which says that an alternative nil-algebra of index \(n\) that is without elements of additive order \(\le n\) is solvable.
Chapter 7 is devoted to establishing the now classical result: a simple alternative algebra is either associative or a Cayley-Dickson algebra over its center.
Then in Chapter 8 the authors take up the study of radicals. After a general introduction to radical theory, the Baer (lower nil), Levitzki (locally nilpotent), Köthe (upper nil), and Andrunakievich (anti-simple) radicals are in turneachextended to the variety of alternative algebras. As it turns out, all four radicals remain hereditary, and the descriptions of the semisimple algebras are analogous to the associative case.
Next the structure of prime and semiprime alternative algebras is investigated in Chapter 9. Except for the characteristic 3 case, which is still open, every prime alternative algebra proves to be either associative or a subring of a specified form in a Cayley-Dickson algebra, i.e. a so called “Cayley-Dickson ring”.
The Jacobson (quasi-regular) radical of an alternative algebra \(A\) is developed in Chapter 10. As in the associative case, \(A\) is called primitive if it has a maximal modular right ideal which does not contain nonzero ideals of \(A\). It is first proved a primitive alternative algebra is either associative or a Cayley-Dickson algebra over its center. Then it is proved that the intersection \(K(A)\) of all maximal modular right ideals of \(A\) defines a hereditary radical such that \(A/K(A)\) is a subdirect sum of primitive alternative algebras. This is followed by the construction of the quasi-regular radical \(S(A)\), and Zhevlakov’s long sought proof that \(S(A) = K(A)\). The chapter then concludes with some results concerning this radical in Cayley-Dickson rings.
In Chapter 11 the concepts of right representation and right module are introduced for algebras from an arbitrary variety. The primitive alternative algebras are then characterized as algebras having faithful irreducible modules, and this in turn is used to characterize the Jacobson radical in terms of representations.
Chapter 12 concerns alternative algebras satisfying various chain conditions on ideals. Among other things, a structure theory analogous to the associative case is obtained for alternative Artinian algebras.
Free alternative algebras are treated in Chapter 13, the longest chapter in the book. The topics covered include identities in finitely-generated algebras, nilpotent elements and radicals of free algebras, and centers of algebras.
In Chapters 14 and 15 the authors return to the study of Jordan algebras. The first of these chapters deals mainly with radicals, especially the McCrimmon and quasi-regular radicals; while Chapter 15 presents a structure theory for Jordan algebras with minimal condition on quadratic ideals.
Chapter 16, the last chapter, is devoted to a basic survey of right alternative algebras.
In addition to the contents outlined above, there are also included 230 exercises, which are distributed fairly evenly throughout. These exercises are not required anywhere in the text, and those that ask the reader to prove results independent of the text are always provided with hints. In the later chapters, the authors also point out several open questions.
Every reader should find this book an excellent as well as long-needed addition to the literature.