The Wedderburn principal theorem for a generalization of alternative algebras

Abstract

A generalized alternative ring I is a nonassociative ring $R$ in which the identities $(wx,y,z) + (w,x,[y,z]) - w(x,y,z) - (w,y,z)x;([w,x],y,z) + (w,x,yz) - y(w,x,z) - (w,x,y)z$; and $(x,x,x)$ are identically zero. It is here demonstrated that if $A$ is a finite-dimensional algebra of this type over a field $F$ of characteristic # 2, 3, then $A$ a nilalgebra implies $A$ is nilpotent. A generalized alternative ring II is a nonassociative ring $R$ in which the identities $(wx,y,z) + (w,x,[y,z]) - w(x,y,z) - (w,y,z)x$ and $(x,y,x)$ are identically zero. Let $A$ be a finite-dimensional algebra of this type over a field $F$ of characteristic # 2. Then it is here established that (1) $A$ a nilalgebra implies $A$ is nilpotent; (2) $A$ simple with no nonzero idempotent other than 1 and $F$ algebraically closed imply $A$ itself is a field; and (3) the standard Wedderburn principal theorem is valid for $A$.