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alternative algebra
A nonassociative algebra $A$ is alternative if
1. (left alternative laws) $[\ a,a,b\ ]=0$, and
2. (right alternative laws) $[\ b,a,a\ ]=0$,
for any $a,b\in A$, where $[\ ,,]$ is the associator on $A$.
Remarks

Let $A$ be alternative and suppose $\operatorname{char}(A)\neq 2$. From the fact that $[\ a+b,a+b,c\ ]=0$, we can deduce that the associator $[\ ,,]$ is anticommutative, when one of the three coordinates is held fixed. That is, for any $a,b,c\in A$,
(a) $[\ a,b,c\ ]=[\ b,a,c\ ]$
(b) $[\ a,b,c\ ]=[\ a,c,b\ ]$
(c) $[\ a,b,c\ ]=[\ c,b,a\ ]$
Put more succinctly,
$[\ a_{1},a_{2},a_{3}\ ]=\operatorname{sgn}(\pi)[\ a_{{\pi(1)}},a_{{\pi(2)}},a_% {{\pi(3)}}\ ],$ where $\pi\in S_{3}$, the symmetric group on three letters, and $\operatorname{sgn}(\pi)$ is the sign of $\pi$.

An alternative algebra is a flexible algebra, provided that the algebra is not Boolean (characteristic $\neq 2$). To see this, replace $c$ in the first anticommutative identities above with $a$ and the result follows.

Artin’s Theorem: If a nonassociative algebra $A$ is not Boolean, then $A$ is alternative iff every subalgebra of $A$ generated by two elements is associative. The proof is clear from the above discussion.

A commutative alternative algebra $A$ is a Jordan algebra. This is true since $a^{2}(ba)=a^{2}(ab)=(ab)a^{2}=((ab)a)a=(a(ab))a=(a^{2}b)a$ shows that the Jordan identity is satisfied.

Alternativity can be defined for a general ring $R$: it is a nonassociative ring such that for any $a,b\in R$, $(aa)b=a(ab)$ and $(ab)b=a(bb)$. Equivalently, an alternative ring is an alternative algebra over $\mathbb{Z}$.
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Comments
characteristic
the word characteristic should link to:
http://planetmath.org/encyclopedia/Characteristic.html
Re: characteristic
please fill a correction
f
G > H G
p \ /_  ~ f(G)
\ / f ker f
G/ker f
Re: characteristic
Thanx drini,
I was thinking about making it corrections instead of posts, but I wasn't sure whether they were indeed that. If they aren't I guess they can be rejected, so I will try to keep the distinction clear,