## You are here

Homepower-associative algebra

## Primary tabs

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

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

- 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, An Introduction on Nonassociative Algebras
^{}, Dover, New York (1995).

## Mathematics Subject Classification

17A05*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections