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.

For a non-associative ring or algebra R, the associator is the multilinear map ${\displaystyle [\cdot ,\cdot ,\cdot ]:R\times R\times R\to R}$ given by

Just as the commutator

measures the degree of non-commutativity, the associator measures the degree of non-associativity of R. 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 R is an alternative ring.

The associator is symmetric in its two rightmost arguments when R 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