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# associator

Let $A$ be a non-associative algebra over a field. The *associator* of $A$, denoted by $[\ ,,]$, is a trilinear map from $A\times A\times A$ to $A$ given by:

$[\ a,b,c\ ]=(ab)c-a(bc).$ |

Just as the commutator measures how close an algebra is to being commutative, the associator measures how close it is to being associative. $[\ ,,]=0$ identically iff $A$ is associative.

# References

- 1 R. D. Schafer, An Introduction on Nonassociative Algebras, Dover, New York (1995).

Defines:

anti-associative

Related:

AlternativeAlgebra, PowerAssociativeAlgebra, FlexibleAlgebra, Commutator

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

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17A01*no label found*

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## Comments

## algebra or ring

couldn't the associater be defined for rings? They seem to have all necessary operations.

## commutator

commutator should point to

http://planetmath.org/encyclopedia/CommutatorLieAlgebra.html

## Re: commutator

HkBSt, the proper way to fix this is not by posting underneath the entry, but filling a correction (the link for that is below the entry when you view it)

the uathor then gets the notice and proceeds to change what's required. So if you find most of these mistakes , please point them to the author using corrections system (besides, that way author gets an email akerting them, in the way you did, will only be fixed if the author looks at the entry)

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f

## Re: algebra or ring

Sure, the associator can be defined for rings. But it is rather useless, since a ring is, by definition, multiplicatively associative and hence, has a trivial associator. When we're speaking of not-necessarily associative algebras, we are talking about a "generalized" ring (over a field) in which the multiplication is not assumed to be associative.

## Re: algebra or ring

But logically it should be defined for something which I will call here a nonassociative ring. The extra operation of scalar multiplication that an algebra has over a ring is superfluous.

Hmmm, it also seems there is no definition of a generalized ring on PM. Maybe I will make an entry sorting this stuff out.

## Re: algebra or ring

You should be careful when using the phrase "nonassociative ring" since it implies the exclusion of any "associative" ring, unless this is exactly your intent. I prefer the naming of a "generalized ring" to include both the traditional "associative" rings as well as the nonassociative ones. I am not aware of any other naming convention for a "generalized" ring in the mathematical community.

But to avoid any more confusion, please add an entry and let's go from there.

Chi