Definitions

# Associator

In abstract algebra, for a ring or algebra $R$, the associator is the multilinear map $R times R times R to R$ given by

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

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

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

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

$a_\left\{x,y,z\right\} : \left(xy\right)z mapsto x\left(yz\right).$