Prime generalized alternative rings $I$ with nontrivial idempotent

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 demonstrated here that if $R$ is a ring of this type with characteristic different from two and three, then $R$ semiprime with idempotent $e$ implies that $R$ has a Peirce decomposition relative to $e$. Furthermore, if $R$ is prime and $e \ne 0,1$; then $R$ must be alternative.