Algebra over a field

[mklabgr]

Algebra over a Field — Summary

An algebra over a field (or $K$-algebra) is a mathematical structure that combines the properties of a vector space with an internal multiplication operation. More precisely, if $K$ is a field, a $K$-algebra $A$ is a vector space over $K$ equipped with a multiplication
$A \times A \to A$
that is bilinear. Thus, for $x,y,z \in A$ and $a,b \in K$,
$(x+y)z=xz+yz,$
$z(x+y)=zx+zy,$
and
$(ax)(by)=ab(xy).$
Importantly, multiplication need not automatically be commutative or associative. Additional assumptions lead to important subclasses such as associative, commutative, and unital algebras.

Many familiar mathematical objects are algebras. The complex numbers $\mathbb{C}$ form a two-dimensional algebra over $\mathbb{R}$; polynomial rings such as $K[x]$ are commutative associative algebras; and the set of $n\times n$ matrices over $K$ forms an associative but generally noncommutative algebra. The vector space $\mathbb{R}^3$, equipped with the vector cross product, gives an example in which multiplication is nonassociative. Other important examples include quaternions, group algebras, function algebras, operator algebras, Lie algebras, Jordan algebras, and octonions.

The theory also introduces algebra homomorphisms, which preserve both the vector-space structure and multiplication; subalgebras, which are vector subspaces closed under multiplication; and ideals, which are subspaces stable under multiplication by elements of the larger algebra. Scalars can also be extended from a field $K$ to a larger field $F$ using the tensor product
$A_F=A\otimes_K F.$
For associative unital algebras, the concept can equivalently be described as a ring $A$ together with a homomorphism
$K\to Z(A),$
where $Z(A)$ denotes the center of the ring.

Key takeaways

• A $K$-algebra is essentially a vector space with a compatible multiplication.
  
• Multiplication is required to be bilinear, but it need not be associative or commutative.
  
• Matrices, polynomials, complex numbers, Lie algebras, and operator algebras are important examples.
  
• The concept provides a bridge between linear algebra and ring theory and is fundamental in abstract algebra, algebraic geometry, representation theory, and functional analysis.
  

ARTICLE