power-associative algebra

Let $A$ be a non-associative algebra. A subalgebra $B$ of $A$ is said to be cyclic if it is generated by one element.

A non-associative algebra is power-associative if, $[B,B,B]=0$ for any cyclic subalgebra $B$ of $A$, where $[-,-,-]$ is the associator.

If we inductively define the powers of an element $a\in A$ by

1. (when $A$ is unital with $1\ne 0$) $a^0:=1$,

2. $a^1:=a$, and

3. $a^n:=a(a^{n-1})$ for $n>1$,

then power-associativity of $A$ means that $[a^i,a^j,a^k]=0$ for any non-negative integers $i,j$ and $k$, since the associator is trilinear (linear in each of the three coordinates). This implies that $a^m{a}^n=a^{m+n}$. In addition, ${(a^m)}^n=a^{mn}$.

A theorem, due to A. Albert, states that any finite power-associative division algebra over the integers of characteristic not equal to 2, 3, or 5 is a field. This is a generalization of the Wedderburn’s Theorem on finite division rings.