Power-associative algebra

Summary

A power-associative algebra is a nonassociative algebra in which the subalgebra generated by any single element is associative. In such an algebra the powers $x^n$ of an element are well defined, whatever the bracketing, and $x^m x^n = x^{m+n}$. Alternative algebras, Jordan algebras and the sedenions are power-associative. It is the natural setting in which one can speak of polynomials in one element, idempotents and nilpotent elements.