# power-associative algebra

Let $A$ be a non-associative algebra. A subalgebra $B$ of $A$ is said to be cyclic if it is generated by one element.

A non-associative algebra is power-associative 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. 1.

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

2. 2.

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

3. 3.

$a^{n}:=a(a^{n-1})$ for $n>1$,

then power-associativity of $A$ means that $[a^{i},a^{j},a^{k}]=0$ for any non-negative 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 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.

## References

• 1 R. D. Schafer, , Dover, New York (1995).
Title power-associative algebra PowerassociativeAlgebra 2013-03-22 14:43:27 2013-03-22 14:43:27 CWoo (3771) CWoo (3771) 15 CWoo (3771) Definition msc 17A05 di-associative diassociative Associator