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In abstract algebra, the term associator is used in different ways as a measure of the nonassociativity of an algebraic structure. Associators are commonly studied as triple systems.

Ring theoryEdit

For a nonassociative ring or algebra  , the associator is the multilinear map   given by


Just as the commutator


measures the degree of noncommutativity, the associator measures the degree of nonassociativity of  . For an associative ring or algebra the associator is identically zero.

The associator in any ring obeys the identity


The associator is alternating precisely when   is an alternative ring.

The associator is symmetric in its two rightmost arguments when   is a pre-Lie algebra.

The nucleus is the set of elements that associate with all others: that is, the n in R such that


The nucleus is an associative subring of R.

Quasigroup theoryEdit

A quasigroup Q is a set with a binary operation   such that for each a,b in Q, the equations   and   have unique solutions x,y in Q. In a quasigroup Q, the associator is the map   defined by the equation


for all a,b,c in Q. As with its ring theory analog, the quasigroup associator is a measure of nonassociativity of Q.

Higher-dimensional algebraEdit

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


Category theoryEdit

In category theory, the associator expresses the associative properties of the internal product functor in monoidal categories.

See alsoEdit


  • Bremner, M.; Hentzel, I. (March 2002). "Identities for the Associator in Alternative Algebras". Journal of Symbolic Computation. 33 (3): 255–273. CiteSeerX doi:10.1006/jsco.2001.0510.
  • Schafer, Richard D. (1995) [1966]. An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5. CS1 maint: discouraged parameter (link)