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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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