Kleinfeld function

Definition

Suppose $R$ is a non-associative ring (i.e., a not necessarily associative ring) with multiplication $*$. Denote by $a$ the associator of $R$, i.e., $a(x,y,z):=((x*y)*z)-(x*(y*z))$. The Kleinfeld function of $R$ is a function $f$ from $R^{4}$ to $R$ defined as follows:

$f(w,x,y,z):=a(w*x,y,z)-(x*a(w,y,z))-(a(x,y,z)*w)$

The Kleinfeld function is typically studied in the context of alternative rings, and it is used in one of the proofs of Artin's theorem on alternative rings.

Facts

• Kleinfeld function is alternating in all pairs of variables for alternative ring