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 bilinear mapping $*$ is often called multiplication.

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

Some sources insist that an algebra over a field requires that the bilinear mapping $*$ must have an identity element $1_A$ such that:

$\forall a \in A: a * 1_A = 1_A * a = a$

that is, that $\struct {A, *}$ has to be a unitary algebra.

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

Especially in commutative algebra, an algebra over a field is often defined as a unital associative commutative algebra.

Some sources refer to an algebra over a field as a linear algebra.

Others call it a hypercomplex number system.

1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): algebra over field
1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem