Definition:Algebra over Field

Definition
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.

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

Also see

 * Definition:Algebra over Ring
 * Definition:Unitary Algebra
 * Definition:Associative Algebra