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In [[abstract algebra]], the term '''associator''' is used in different ways as a measure of the [[associativity|nonassociativity]] of an algebraic structure.
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==Ring theory==
 
For a [[nonassociative ring]] or [[Algebra over a field#Non-associative algebras|algebra]] <math>R</math>, the '''associator''' is the [[multilinear map]] <math>[\cdot,\cdot,\cdot] : R \times R \times R \to R</math> given by
 
:<math>[x,y,z] = (xy)z - x(yz).\,</math>
 
Just as the [[commutator]] measures the degree of [[commutativity|noncommutativity]], the associator measures the degree of nonassociativity of <math>R</math>.
It is identically zero for an [[associative ring]] or algebra.
 
The associator in any ring obeys the identity
 
:<math>w[x,y,z] + [w,x,y]z = [wx,y,z] - [w,xy,z] + [w,x,yz].\,</math>
 
The associator is [[alternating form|alternating]] precisely when <math>R</math> is an [[alternative ring]].
 
The associator is symmetric in its two rightmost arguments when <math>R</math> 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
 
:<math> [n,R,R] = [R,n,R] = [R,R,n] = \{0\} \ . </math>
 
==Quasigroup theory==
 
A [[quasigroup]] ''Q'' is a set with a binary operation <math>\cdot : Q\times Q\to Q</math> such that for each ''a,b'' in ''Q'',
the equations <math>a\cdot x = b</math> and <math>y\cdot a = b</math> have unique solutions ''x,y'' in ''Q''. In a quasigroup ''Q'', the  
associator is the map <math>(\cdot,\cdot,\cdot) : Q\times Q\times Q\to Q</math> defined by the equation
 
:<math>(a\cdot b)\cdot c=(a\cdot (b\cdot c))\cdot (a,b,c)</math>
 
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
 
:<math> a_{x,y,z} : (xy)z \mapsto x(yz).</math>
 
==Category theory==
In [[category theory]], the associator expresses the associative properties of the internal product functor in [[monoidal category|monoidal categories]].
 
== See also ==
 
*[[Commutator]]
 
{{algebra-stub}}
[[Category:Non-associative algebra]]
==References==
* {{cite web | title=Identities for the Associator in Alternative Algebras | author=Bremner and Hentzel | url=http://math.usask.ca/~bremner/research/publications/BHiaaa.pdf }}
* {{cite book |first=Richard D. |last=Schafer |title=An Introduction to Nonassociative Algebras |year=1995 |origyear=1966 |publisher=Dover |isbn=0-486-68813-5 }}

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