Algebra over a field

In mathematics, an algebra over a field (often simply called an algebra) is a special kind of math structure. It is a vector space with a way to multiply its parts together. This multiplication follows certain rules, which makes algebra useful in many areas of math.

One feature of an algebra is that its multiplication might or might not follow the associative property. When it does, it is called an associative algebra. For example, the group of real square matrices of a certain size forms an associative algebra over the field of real numbers. These matrices can be added and multiplied together in a way that keeps the structure consistent.

Algebras can also have a special element called an identity element, which works like the number 1 in normal multiplication. When an algebra has this element, it is called unital or unitary. The ring of real square matrices has this property because there is a special matrix, the identity matrix, that does not change other matrices when multiplied by them. This makes it a unital associative algebra, meaning it is both associative and has an identity element.

Definition and motivation

An algebra over a field is a special kind of mathematical space. It mixes ideas from two things: vector spaces and multiplication.

A vector space is a group of objects you can add together. You can also stretch or shrink these objects by numbers, called scalars.

In an algebra, we also have a rule for multiplication. This rule tells us how to combine any two objects to get another object.

The multiplication follows certain rules, like how we multiply numbers normally. For example, multiplying a sum of two objects by another object should give the same result as multiplying each object separately and then adding those results.

Algebras can be associative, meaning the order of multiplication does not matter. Or they can be non-associative, meaning the order does matter.


Algebra	vector space	bilinear operator	associativity	commutativity
complex numbers	R 2 {\displaystyle \mathbb {R} ^{2}}	product of complex numbers ( a + i b ) ⋅ ( c + i d ) {\displaystyle \left(a+ib\right)\cdot \left(c+id\right)}	yes	yes
cross product of 3D vectors	R 3 {\displaystyle \mathbb {R} ^{3}}	cross product a → × b → {\displaystyle {\vec {a}}\times {\vec {b}}}	no	no (anticommutative)
quaternions	R 4 {\displaystyle \mathbb {R} ^{4}}	Hamilton product ( a + v → ) ( b + w → ) {\displaystyle (a+{\vec {v}})(b+{\vec {w}})}	yes	no
polynomials	R [ X ] {\displaystyle \mathbb {R} [X]}	polynomial multiplication	yes	yes
square matrices	R n × n {\displaystyle \mathbb {R} ^{n\times n}}	matrix multiplication	yes	no

Basic concepts

An algebra over a field is a special mathematical space. It has two main actions: addition and multiplication. We can also multiply by numbers from a field, which changes the space in certain ways.

We can study smaller parts of an algebra called subalgebras. These are groups of elements inside the algebra that stay the same after addition, multiplication, and multiplying by numbers. We can also look at special groups called ideals. These behave in special ways when multiplied by other elements in the algebra.

Kinds of algebras and examples

Algebras over fields can be many different types, depending on extra rules we add, like whether their multiplication is commutative or associative. These rules change how we study them.

One special type is a unital algebra, which has a special element that acts like a "1", so when you multiply anything by this element, you get the same thing back. Another type is a zero algebra, where multiplying any two elements always gives zero.

An associative algebra follows the usual order of operations when multiplying. Examples include matrices and polynomials.

A non-associative algebra doesn’t need to follow that order. Examples include special kinds of numbers like octonions and structures like Euclidean space.

Main article: Associative algebra

Main article: Non-associative algebra

Algebras and rings

An algebra over a field is a special type of ring. This ring connects to the field in a way that keeps things working well. This connection makes sure the algebra works nicely with the ring and the field.

When we have two algebras, a special map called a homomorphism can be used between them. This map keeps the ring structure and the field's actions the same. It means the map respects how elements are multiplied and added in both algebras.

Structure coefficients

Main article: Structure constants

In algebras over a field, the way elements multiply together is decided by how the basic building blocks, or basis elements, multiply. Once we pick these building blocks, we can decide how they multiply, and this decision tells us how all other elements multiply.

For a finite-dimensional algebra, this multiplication is captured by special numbers called structure coefficients. If the algebra has dimension n, we need n³ such coefficients, written c_i,j,k, to fully describe the multiplication. These coefficients follow a rule that shows how the basis elements combine to form other elements in the algebra.

Classification of low-dimensional unital associative algebras over the complex numbers

Two-dimensional, three-dimensional, and four-dimensional algebras over complex numbers have been fully classified.

In two dimensions, there are two types of algebras. Each uses two basic elements: 1 (which acts like the number 1) and another element called a.

For three-dimensional algebras, there are five different types. Each uses three basic elements: 1, a, and b. The way these elements multiply together determines which type of algebra it is. One of these five types does not follow the usual order of multiplication, while the others do.

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Generalization: algebra over a ring

In some parts of mathematics, like commutative algebra, we talk about something called an algebra over a ring. Instead of using a field, we use a special kind of mathematical object called a commutative ring. The main idea stays the same, but we change one part: we think of the algebra as a special kind of structure called an R-module instead of a vector space.

A ring can always be seen as an associative algebra over its center and over the integers.