Definition:Algebra over Field

Definition
An algebra over a field $\left({G_F, \oplus}\right)$ is a vector space $G_F$ over a field $F$ with a bilinear mapping $\oplus: G^2 \to G$.

That is, it is an algebra over a ring $\left({G_R, \oplus}\right)$ where the ring $R$ is a field, and the $R$-module $G_R$ is a vector space.

The symbol $A$ is often used for such an algebra, more so as the level of abstraction increases.

Also see

 * Algebra over a Ring