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.
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.
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.
Sources
- 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
- 2002: John C. Baez: The Octonions: 1.1 Preliminaries
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): algebra: 2.