Associator

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 theory.
For a nonassociative ring or algebra formula_1, the associator is the multilinear map formula_2 given by
Just as the commutator 
measures the degree of noncommutativity, the associator measures the degree of nonassociativity of formula_1.
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 formula_1 is an alternative ring.
The associator is symmetric in its two rightmost arguments when formula_1 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 formula_10 such that for each "a,b" in "Q",
the equations formula_11 and formula_12 have unique solutions "x,y" in "Q". In a quasigroup "Q", the 
associator is the map formula_13 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.