Free alternative algebras of rank 3
type: Item
instance of: scholarly article
title: Free alternative algebras of rank 3
zbMATH Open document ID: 0588.17020
DOI: 10.1007/BF01979703
author: A. V. Il'tyakov
published in: Algebra and Logic
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/187260
review text: The author shows that the free alternative algebra A of rank 3 over a field F of characteristic \(\neq 2,3\) does not contain nilpotent elements. Let A be a free alternative algebra of rank 3 over a field F of characteristic \(\neq 2,3\). Let N(A) be the nucleus of A, and D(A) be the ideal generated by all associators. Let T(C) be the ideal of homogeneous identities for a splittable Cayley-Dickson algebra. The author proves that \(T(C)\cap D(A)=0\) by showing that a basis of \(D(A)/(T(C)\cap D(A))\) is actually a basis of D(A). Since the author has previously proven that the product of two elements of A is zero if and only if one of the elements belongs to D(A) and the other belongs to T(C), this proves the theorem.
Mathematics Subject Classification ID: 17D05
Mathematics Subject Classification ID: 17D10
zbMATH DE Number: 3944105
zbMATH Keywords: characteristic not two or three
zbMATH Keywords: free alternative algebra
zbMATH Keywords: nilpotent elements
zbMATH Keywords: Cayley-Dickson algebra
reviewed by: Irvin Roy Hentzel
MaRDI profile type: Publication
cites work: On free alternative rings
cites work: Malcev Algebras
description: scientific article; zbMATH DE number 3944105