In abstract algebra, the term associator is used in different ways as a measure of the non-associativity of an algebraic structure. Associators are commonly studied as triple systems. Ring theory For a non-associative ring or algebra , the associator is the multilinear map given by Just as the commutator measures the degree of non-commutativity, the associator measures the degree of non-associativity 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 theory 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 algebra In higher-dimensional algebra, where there may be non-identity morphisms between algebraic expressions, an associator is an isomorphism Category theory In category theory, the associator expresses the associative properties of the internal product functor in monoidal categories. See also Commutator Non-associative algebra Quasi-bialgebra – discusses the Drinfeld associator References Non-associative algebra