Alternative algebras and RA loops

Abstract

In the first part of this thesis, we study the relationships between three algebra structures: Cayley-Dickson algebras, RA loops and alternative loop algebras. -- Let R be a commutative associative ring with 1 and let A be an R-algebra with unity of characteristic different from 2. For any α, β and γ ∈ A, let A(α,β,γ) be the Cayley-Dickson algebra. We construct an RA loop L from each Cayley-Dickson algebra A(α,β,γ), called the induced RA loop. We show that any RA loop is a homomorphic image of some induced RA loop. After introducing the category of Cayley-Dickson algebras and the category of RA loops, we show that the two categories are equivalent. -- Using the induced RA loops, we show that any Cayley-Dickson algebra is a homomorphic image of an alternative loop algebra. Thus we give a new way of representing a Cayley-Dickson algebra. Furthermore, the homomorphism commutes with the norm and trace operations of the alternative loop algebra and the Cayley-Dickson algebra. The kernel of this homomorphism is completely determined. The prime radical and Jacobson radical of some Cayley-Dickson algebras are determined. A result of de Barros is generalized. The more general form of the homomorphism is studied. -- Necessary and sufficient conditions for an RA loop to be the Moufang circle loop of a quasiregular alternative algebra are given. The algebra structure of a finite alternative nilpotent ring with the Moufang circle loop being an RA loop is completely determined. -- In the second part of this thesis, the alternative rings of order p⁴ and p⁵ are completely determined, where p is a prime. This generalizes a result of A. T. Gainov. The two smallest alternative rings have order 2⁴ . For each prime number, there are fifteen alternative rings of order pn, n ≤ 5. The relationships between these fifteen rings are described. From these alternative algebras, a class of group-graded alternative algebras is derived.