Algebra over a field

In mathematics, an algebra over a field is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear".
The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras where associativity of multiplication is assumed, and non-associative algebras, where associativity is not assumed. Given an integer, the ring of real square matrices of order is an example of an associative algebra over the field of real numbers under matrix addition and matrix multiplication since matrix multiplication is associative. Three-dimensional Euclidean space with multiplication given by the vector cross product is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the Jacobi identity instead.
An algebra is unital or unitary if it has an identity element with respect to the multiplication. The ring of real square matrices of order forms a unital algebra since the identity matrix of order is the identity element with respect to matrix multiplication. It is an example of a unital associative algebra, a ring that is also a vector space.
Many authors use the term algebra to mean associative algebra, or unital associative algebra, or in some subjects such as algebraic geometry, unital associative commutative algebra.
Replacing the field of scalars by a commutative ring leads to the more general notion of an algebra over a ring. Algebras are not to be confused with vector spaces equipped with a bilinear form, like inner product spaces, as, for such a space, the result of a product is not in the space, but rather in the field of coefficients.

Definition and motivation

Motivating examples

Definition

Let be a field, and let be a vector space over equipped with an additional binary operation from to, denoted here by . Then is an algebra over if the following identities hold for all elements in, and all elements and in :

Right distributivity:
Left distributivity:
Compatibility with scalars:.

These three conditions are another way of saying that the binary operation is bilinear. An algebra over is sometimes also called a -algebra, and is called the base field of. The binary operation is often referred to as multiplication in. The convention adopted in this article is that multiplication of elements of an algebra is not necessarily associative, although some authors use the term algebra to refer to an associative algebra.
When a binary operation on a vector space is commutative, left distributivity and right distributivity are equivalent, and, in this case, only one distributivity requires a proof. In general, for non-commutative operations left distributivity and right distributivity are not equivalent, and require separate proofs.

Basic concepts

Algebra homomorphisms

Given -algebras and, a homomorphism of -algebras or -algebra homomorphism is a -linear map such that for all in. If and are unital, then a homomorphism satisfying is said to be a unital homomorphism. The space of all -algebra homomorphisms between and is frequently written as
A -algebra isomorphism is a bijective -algebra homomorphism.

Subalgebras and ideals

A subalgebra of an algebra over a field is a linear subspace that has the property that the product of any two of its elements is again in the subspace. In other words, a subalgebra of an algebra is a non-empty subset of elements that is closed under addition, multiplication, and scalar multiplication. In symbols, we say that a subset of a -algebra is a subalgebra if for every, in and in, we have that,, and are all in.
In the above example of the complex numbers viewed as a two-dimensional algebra over the real numbers, the one-dimensional real line is a subalgebra.
A left ideal of a -algebra is a linear subspace that has the property that any element of the subspace multiplied on the left by any element of the algebra produces an element of the subspace. In symbols, we say that a subset of a -algebra is a left ideal if for every and in, in and in, we have the following three statements.

is in ,
is in ,
is in .

If were replaced with is in, then this would define a right ideal. A two-sided ideal is a subset that is both a left and a right ideal. The term ideal on its own is usually taken to mean a two-sided ideal. Of course when the algebra is commutative, then all of these notions of ideal are equivalent. Conditions and together are equivalent to being a linear subspace of. It follows from condition that every left or right ideal is a subalgebra.
This definition is different from the definition of an ideal of a ring, in that here we require the condition. Of course if the algebra is unital, then condition implies condition.

Extension of scalars

If we have a field extension, which is to say a bigger field that contains, then there is a natural way to construct an algebra over from any algebra over. It is the same construction one uses to make a vector space over a bigger field, namely the tensor product. So if is an algebra over, then is an algebra over.

Kinds of algebras and examples

Algebras over fields come in many different types. These types are specified by insisting on some further axioms, such as commutativity or associativity of the multiplication operation, which are not required in the broad definition of an algebra. The theories corresponding to the different types of algebras are often very different.

Unital algebra

An algebra is unital or unitary if it has a unit or identity element with for all in the algebra.

Zero algebra

An algebra is called a zero algebra if for all, in the algebra, not to be confused with the algebra with one element. It is inherently non-unital, associative and commutative.
A unital zero algebra is the direct sum of a field and a -vector space, that is equipped by the only multiplication that is zero on the vector space, and makes it an unital algebra.
More precisely, every element of the algebra may be uniquely written as with and, and the product is the only bilinear operation such that for every and in. So, if and, one has
A classical example of unital zero algebra is the algebra of dual numbers, the unital zero -algebra built from a one dimensional real vector space.
This definition extends verbatim to the definition of a unital zero algebra over a commutative ring, with the replacement of "field" and "vector space" with "commutative ring" and "module".
Unital zero algebras allow the unification of the theory of submodules of a given module and the theory of ideals of a unital algebra. Indeed, the submodules of a module correspond exactly to the ideals of that are contained in.
For example, the theory of Gröbner bases was introduced by Bruno Buchberger for ideals in a polynomial ring over a field. The construction of the unital zero algebra over a free -module allows extending this theory as a Gröbner basis theory for submodules of a free module. This extension allows, for computing a Gröbner basis of a submodule, to use, without any modification, any algorithm and any software for computing Gröbner bases of ideals.
Similarly, unital zero algebras allow to deduce straightforwardly the Lasker–Noether theorem for modules from the original Lasker–Noether theorem for ideals.

Associative algebra

Examples of associative algebras include

the algebra of all -by- matrices over a field . Here the multiplication is ordinary matrix multiplication.
group algebras, where a group serves as a basis of the vector space and algebra multiplication extends group multiplication.
the commutative algebra of all polynomials over .
algebras of functions, such as the -algebra of all real-valued continuous functions defined on the interval, or the -algebra of all holomorphic functions defined on some fixed open set in the complex plane. These are also commutative.
Incidence algebras are built on certain partially ordered sets.
algebras of linear operators, for example on a Hilbert space. Here the algebra multiplication is given by the composition of operators. These algebras also carry a topology; many of them are defined on an underlying Banach space, which turns them into Banach algebras. If an involution is given as well, we obtain B*-algebras and C*-algebras. These are studied in functional analysis.
Non-associative algebra

A non-associative algebra over a field is a -vector space A equipped with a -bilinear map. The usage of "non-associative" here is meant to convey that associativity is not assumed, but it does not mean it is prohibited – that is, it means "not necessarily associative".
Examples detailed in the main article include:

Euclidean space with multiplication given by the vector cross product
Octonions
Lie algebras
Jordan algebras
Alternative algebras
Flexible algebras
Power-associative algebras
Algebras and rings

The definition of an associative -algebra with unit is also frequently given in an alternative way. In this case, an algebra over a field is a ring together with a ring homomorphism
where is the center of. Since is a ring homomorphism, then one must have either that is the zero ring, or that is injective. This definition is equivalent to that above, with scalar multiplication
given by
Given two such associative unital -algebras and, a unital -algebra homomorphism is a ring homomorphism that commutes with the scalar multiplication defined by, which one may write as
for all and. In other words, the following diagram commutes: