Artin's Theorem on Alternative Algebras
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Theorem
Let $A = \struct {A_R, \oplus}$ be an algebra over the ring $R$ such that $A$ is not a boolean algebra.
Then $A$ is alternative if and only if:
- $\forall a, b \in A: \paren {a \oplus a} \oplus b = a \oplus \paren {a \oplus b}$
- $\forall a, b \in A: \paren {b \oplus a} \oplus a = b \oplus \paren {a \oplus a}$
Alternative Definition
Some sources take this as a definition of an alternative algebra and from it deduce that $A$ is alternative if and only if:
- for all $a, b \in A_R$, the subalgebra generated by $\set {a, b}$ is an associative algebra.
However, the latter statement is how an alternative algebra is defined on this site, and from it we deduce the statements given in the exposition of this theorem.
The approaches are clearly equivalent.
Proof
When $A$ is
So suppose that:
- $\forall a, b \in A: \paren {a \oplus a} \oplus b = a \oplus \paren {a \oplus b}$
- $\forall a, b \in A: \paren {b \oplus a} \oplus a = b \oplus \paren {a \oplus a}$
Then:
- $\sqbrk {a, a, b} = 0$
- $\sqbrk {b, a, a} = 0$
where $\sqbrk {a, a, b}$ denotes the associator of $a, b \in A_R$.
Now let us compute, using the linearity of $\oplus$ and the two suppositions:
\(\ds \paren {a - b} \oplus \paren {\paren {a - b} \oplus a}\) | \(=\) | \(\ds a \oplus \paren {\paren {a - b} \oplus a} - b \oplus \paren {\paren {a - b} \oplus a}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a \oplus \paren {a \oplus a} - a \oplus \paren {b \oplus a} - b \oplus \paren {a \oplus a} + b \oplus \paren {b \oplus a}\) | ||||||||||||
\(\ds \paren {\paren {a - b} \oplus \paren {a - b} } \oplus a\) | \(=\) | \(\ds \paren {a \oplus a} \oplus a - \paren {a \oplus b} \oplus a - \paren {b \oplus a} \oplus a + \paren {b \oplus b} \oplus a\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 0\) | \(=\) | \(\ds \sqbrk {a - b, a - b, a}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqbrk {a, a, a} - \sqbrk {a, b, a} - \sqbrk {b, a, a} + \sqbrk {b, b, a}\) | Subtract the two expressions | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqbrk {a, b, a}\) |
This article, or a section of it, needs explaining. In particular: This suffices, but why it does could do with explanation (although trivial). Also, where do I use that $A$ isn't Boolean? Maybe better use talk page You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
$\blacksquare$
Source of Name
This entry was named for Emil Artin.
Sources
- This article incorporates material from alternative algebra on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.