power-associative algebraPowerAssociativeAlgebra2004-10-10 14:29:382008-12-11 15:09:43Definition% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet $A$ be a non-associative algebra. A subalgebra $B$ of $A$ is said to be \emph{cyclic} if it is generated by one element.
A non-associative algebra is \emph{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
\begin{enumerate}
\item (when $A$ is unital with $1\neq0$) $a^0:=1$,
\item $a^1:=a$, and
\item $a^n:=a(a^{n-1})$ for $n>1$,
\end{enumerate}
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 $a^ma^n=a^{m+n}$. 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.
\begin{thebibliography}{8}
\bibitem{Shafer} R. D. Schafer, {\em An Introduction on Nonassociative Algebras}, Dover, New York (1995).
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