\begin{definition}[Definition:Algebra over Field]
Let $F$ be a field.
An '''algebra over $F$''' is an ordered pair $\struct {A, *}$ where:
:$A$ is a vector space over $F$
:$* : A^2 \to A$ is a bilinear mapping
That is, it is an algebra $\struct {A, *}$ over the ring $F$ where:
:$F$ is a field
:the $F$-module $A$ is a vector space.
\end{definition}