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 $\oplus: G_F^2 \to G_F$.

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 bilinear mapping $\oplus$ is often called multiplication.

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

Variants
Some sources insist that an algebra over a field requires that the bilinear mapping $\oplus$ must have an identity element $1_G$ such that:
 * $\forall a \in G_R: a \oplus 1_G = 1_G \oplus a = a$

... that is, that $\left({G_F, \oplus}\right)$ has to be a unitary algebra.

It is worth being certain of what is meant in any works read.

Also see

 * Algebra over a Ring