powerassociative algebra
Let $A$ be a nonassociative algebra. A subalgebra^{} $B$ of $A$ is said to be cyclic if it is generated by one element.
A nonassociative algebra is powerassociative if, $[B,B,B]=0$ for any cyclic subalgebra $B$ of $A$, where $[,,]$ is the associator^{}.
If we inductively define the powers of an element $a\in A$ by

1.
(when $A$ is unital with $1\ne 0$) ${a}^{0}:=1$,

2.
${a}^{1}:=a$, and

3.
${a}^{n}:=a({a}^{n1})$ for $n>1$,
then powerassociativity of $A$ means that $[{a}^{i},{a}^{j},{a}^{k}]=0$ for any nonnegative integers $i,j$ and $k$, since the associator is trilinear (linear in each of the three coordinates). This implies that ${a}^{m}{a}^{n}={a}^{m+n}$. In addition, ${({a}^{m})}^{n}={a}^{mn}$.
A theorem, due to A. Albert, states that any finite powerassociative 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.
References
 1 R. D. Schafer, An Introduction on Nonassociative Algebras, Dover, New York (1995).
Title  powerassociative algebra 

Canonical name  PowerassociativeAlgebra 
Date of creation  20130322 14:43:27 
Last modified on  20130322 14:43:27 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  15 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 17A05 
Synonym  diassociative 
Synonym  diassociative 
Related topic  Associator 