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power-associative algebra
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(Definition)
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A non-associative algebra is power-associative if, for any cyclic subalgebra
 , ![$ [B,B,B]=0$](http://images.planetmath.org:8080/cache/objects/6350/l2h/img2.png "$ [B,B,B]=0$") , where ![$ [-,-,-]$](http://images.planetmath.org:8080/cache/objects/6350/l2h/img3.png "$ [-,-,-]$") is the associator.
If we inductively define the powers of an element  by
- (when
 is unital with  )  ,
 , and-
 for  ,
then power-associativity of  means that
![$ [a^i,a^j,a^k]=0$](http://images.planetmath.org:8080/cache/objects/6350/l2h/img12.png "$ [a^i,a^j,a^k]=0$") for any non-negative integers  and  , since the associator is trilinear (linear in each of the three coordinates). This implies that
 . In addition,
 .
A theorem, due to A. Albert, states that any finite power-associative division algebra over the integers of characteristic not equal to 2, 3, or 5 is a field. This is a generalization of the Wedderburn's Theorem on finite division rings.
- 1
- R. D. Schafer, An Introduction on Nonassociative Algebras, Dover, New York (1995).
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"power-associative algebra" is owned by CWoo.
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(view preamble)
Cross-references: division rings, Wedderburn's theorem, field, characteristic, division algebra, finite, addition, implies, coordinates, integers, unital, powers, associator, cyclic, non-associative algebra
There is 1 reference to this entry.
This is version 10 of power-associative algebra, born on 2004-10-10, modified 2006-01-30.
Object id is 6350, canonical name is PowerAssociativeAlgebra.
Accessed 1652 times total.
Classification:
| AMS MSC: | 17A05 (Nonassociative rings and algebras :: General nonassociative rings :: Power-associative rings) |
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Pending Errata and Addenda
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