Power-associative algebra

Remarks

Equivalently, any two products of $n$ copies of $x$, with any bracketing, are equal for every $n$; their common value is written $x^n$, and then $x^m x^n = x^{m+n}$. Example: every alternative algebra is power-associative, by Artin's theorem. Example: an anticommutative algebra is power-associative, since $xx = 0$ makes every product of two or more copies of $x$ zero. Non-example: let $A$ have basis $e, f$ over a field, with $ee = f$, $ef = e$ and $fe = ff = 0$. Then $(ee)e = 0$ and $e(ee) = e$, so the subalgebra generated by $e$ is not associative.