Definitions

# Example of a non-associative algebra

This page presents and discusses an example of a non-associative division algebra over the real numbers.

The multiplication is defined by taking the complex conjugate of the usual multiplication: $a*b=overline\left\{ab\right\}$. This is a commutative, non-associative division algebra of dimension 2 over the reals, and has no unit element.

## Proof that $\left(mathbb\left\{C\right\},*\right)$ is a division algebra

For a proof that $mathbb\left\{R\right\}$ is a field, see real number. Then, the complex numbers themselves clearly form a vector space.

It remains to prove that the binary operation given above satisfies the requirements of a division algebra

• (x + y)z = x z + y z;
• x(y + z) = x y + x z;
• (a x)y = a(x y); and
• x(b y) = b(x y);

for all scalars a and b in $mathbb\left\{R\right\}$ and all vectors x, y, and z (also in $mathbb\left\{C\right\}$).

For distributivity:

$x*\left(y+z\right)=overline\left\{x\left(y+z\right)\right\}=overline\left\{xy+xz\right\}=overline\left\{xy\right\}+overline\left\{xz\right\}=x*y+x*z$.

(similarly for right distributivity); and for the third and fourth requirements

$\left(ax\right)*y=overline\left\{\left(ax\right)y\right\}=overline\left\{a\left(xy\right)\right\}=overline\left\{a\right\}cdotoverline\left\{xy\right\}=overline\left\{a\right\}\left(x*y\right)$.

## Non associativity of $\left(mathbb\left\{C\right\},*\right)$

• :$a * \left(b * c\right) = a * overline\left\{b c\right\} = overline\left\{a overline\left\{b c\right\}\right\} = overline\left\{a\right\} b c$
• :$\left(a * b\right) * c = overline\left\{a b\right\} * c = overline\left\{overline\left\{a b\right\} c\right\} = a b overline\left\{c\right\}$

So, if a, b, and c are all non-zero, and if a and c do not differ by a real multiple, $a * \left(b * c\right) neq \left(a * b\right) * c$.

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