Definition:Algebra over Field

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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.


Multiplication

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.


Examples

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.


$2 \times 2$ Matrices under Multiplication

The set of all $2 \times 2$ matrices with real or complex entries forms an algebra over the field of real numbers.


Order $n$ Real Square Matrices under Multiplication

The set of all order $n$ square matrices with real entries forms an algebra over the field of real numbers.


Also known as

Other terms for an algebra over a field include:

a linear algebra
a hypercomplex number system
an associative algebra, but that term is usually applied to something more specific.


Also see

  • Results about algebras over fields can be found here.


Sources

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