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associator ：ウィキペディア英語版

associator
In abstract algebra, the term associator is used in different ways as a measure of the nonassociativity of an algebraic structure.
==Ring theory==

For a nonassociative ring or algebra

R

, the associator is the multilinear map

) : R \times R \times R \to R

given by
:

) = (xy)z - x(yz).\,

Just as the commutator measures the degree of noncommutativity, the associator measures the degree of nonassociativity of

R

.
It is identically zero for an associative ring or algebra.
The associator in any ring obeys the identity
:

w() + ()z = () - () + ().\,

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
:

() = () = () = \ \ .

It turns out that any two of

((),() , ())

being

\

implies that the third is also the zero set.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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