$[x,y,z]\; =\; (xy)z\; -\; x(yz).,$

Just like the commutator measures the degree of noncommutativity, the associator measures the degree of nonassociativity of a nonassociative ring or algebra. It is identically zero for an associative ring or algebra.

The associator in any ring obeys the identity

$w[x,y,z]\; +\; [w,x,y]z\; =\; [wx,y,z]\; -\; [w,xy,z]\; +\; [w,x,yz].,$

The associator is alternating precisely when $R$ is an alternative ring.

In higher-dimensional algebra, where there may be non-identity morphisms between algebraic expressions, an associator is an isomorphism

$a\_\{x,y,z\}\; :\; (xy)z\; mapsto\; x(yz).$

See also

• commutator