Simple (γ, δ) algebras are associative

AbstractA (γ, δ) algebra over a field F is a nonassociative algebra satisfying an identity of the form, (a, b, c) + γ(b, a, c) + δ(c, a, b) = 0, for fixed γ, δ ϵ F, and γ2 − δ2 + δ = 1. We assume that F is of characteristic ≠ 2, ≠ 3; however, we do not assume that the algebra is finite-dimensional over F. We show that any simple (γ, δ) algebra is associative with the possible exception of the cases (± 1, 0) and (1, 1). The approach used in this paper is to represent the identities by matrices by way of the group algebra representation. This enables us to manipulate identities by the well-known techniques of matrix theory

Simple (γ, δ) algebras are associative

AbstractA (γ, δ) algebra over a field F is a nonassociative algebra satisfying an identity of the form, (a, b, c) + γ(b, a, c) + δ(c, a, b) = 0, for fixed γ, δ ϵ F, and γ2 − δ2 + δ = 1. We assume that F is of characteristic ≠ 2, ≠ 3; however, we do not assume that the algebra is finite-dimensional over F. We show that any simple (γ, δ) algebra is associative with the possible exception of the cases (± 1, 0) and (1, 1). The approach used in this paper is to represent the identities by matrices by way of the group algebra representation. This enables us to manipulate identities by the well-known techniques of matrix theory

On Prime Right Alternative Algebras and Alternators

We study subvarieties of the variety of right alternative algebras over a field of characteristic t2,t3 such that the defining identities of the variety force the span of the alternators to be an ideal and do not force an algebra with identity element to be alternative. We call a member of such a variety a right alternative alternator ideal algebra. We characterize the algebras of this subvariety by finding an identity which holds if and only if an algebra belongs to the subvariety. We use this identity to prove that if R is a prime, right alternative alternator ideal algebra with an idempotent e to,tl such that (e,e,R) =O, then either R is alternative or R belongs to one of four exceptional varieties

On Prime Right Alternative Algebras and Alternators

We study subvarieties of the variety of right alternative algebras over a field of characteristic t2,t3 such that the defining identities of the variety force the span of the alternators to be an ideal and do not force an algebra with identity element to be alternative. We call a member of such a variety a right alternative alternator ideal algebra. We characterize the algebras of this subvariety by finding an identity which holds if and only if an algebra belongs to the subvariety. We use this identity to prove that if R is a prime, right alternative alternator ideal algebra with an idempotent e to,tl such that (e,e,R) =O, then either R is alternative or R belongs to one of four exceptional varieties.This article is published as Hentzel, Irvin Roy and Giulia Maria Piacentini Cattaneo “On Prime Right Alternative Algebras and Alternators,” Mathematical Reports – Comptes rendus mathématiques, v. 6, no.1 (1984): 3-8. Posted with permission.</p