Identity of an algebra

Summary

An identity of a nonassociative algebra is a nonassociative polynomial that vanishes when its variables are replaced by any elements of the algebra. The commutative law $xy - yx$, the associative law $(xy)z - x(yz)$, the Jacobi identity and the Jordan identity are examples. Classes of algebras such as associative, alternative, Lie and Jordan algebras are defined by identities; each such class is a variety of algebras. For associative algebras, the theory of identities is the theory of polynomial identity rings.