Definition:Algebra over Field

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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.


The bilinear mapping $*$ is often referred to as multiplication.

Also defined as

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.


Vectors in $3$-Space with Cross Product

Let $V$ be the vector space formed of the set of all vectors in space.

Then $\struct {V, \times}$ forms an algebra over the field of vectors in space where $\times$ is the vector cross product.

Also known as

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

Others call it a hypercomplex number system.

Also see

  • Results about algebras over fields can be found here.