Definition 1

An algebra $\mathbf{F}$ is called alternative if it satisfies the equivalent conditions of Proposition 1.

An associative algebra is obviously alternative. In no. 3 we shall give an example of an alternative algebra which is not associative.

If $\mathbf{F}$ is an alternative A-algebra, every A-algebra $\mathbf{F} \otimes_A A'$ obtained from $\mathbf{F}$ by extending the scalars (§ 1, no. 5) is an alternative $A'$-algebra, as follows from condition (b) of Proposition 1.