In abstract algebra, for a ring or algebra R, the associator is the multilinear map R imes R imes R o R given by

: [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


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