Just as the commutator
The associator in any ring obeys the identity
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.
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.
In higher-dimensional algebra, where there may be non-identity morphisms between algebraic expressions, an associator is an isomorphism
- Bremner, M.; Hentzel, I. (March 2002). "Identities for the Associator in Alternative Algebras". Journal of Symbolic Computation. 33 (3): 255–273. CiteSeerX 10.1.1.85.1905. doi:10.1006/jsco.2001.0510.
- Schafer, Richard D. (1995) . An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5. CS1 maint: discouraged parameter (link)