Definition:Alternative Algebra

Definition

Let $\left({A_R, \oplus}\right)$ be an algebra over a ring $R$.

Then $\left({A_R, \oplus}\right)$ is an alternative algebra if and only if $\oplus$ is alternative.

That is:

For all $a, b \in A_R$, the subalgebra generated by $\left\{{a, b}\right\}$ is an associative algebra.