Free alternative algebras of rank 3
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.
scientific article; zbMATH DE number 3944105