Algebra over a field

In, an algebra over a field (often simply called an algebra) is a equipped with a. Thus, an algebra is an, which consists of a , together with operations of multiplication, addition, and by elements of the underlying , and satisfies the axioms implied by "vector space" and "bilinear".

The multiplication operation in an algebra may or may not be, leading to the notions of s and s. Given an integer n, the of   of order n is an example of an associative algebra over the field of s under  and  since matrix multiplication is associative. Three-dimensional with multiplication given by the  is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the  instead.

An algebra is unital or unitary if it has an with respect to the multiplication. The ring of real square matrices of order n forms a unital algebra since the of order n is the identity element with respect to matrix multiplication. It is an example of a unital associative algebra, a 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, unital associative commutative algebra.

Replacing the field of scalars by a leads to the more general notion of an. Algebras are not to be confused with vector spaces equipped with a, like s, as, for such a space, the result of a product is not in the space, but rather in the field of coefficients.

First example: The complex numbers
Any may be written a + bi, where a and b are s and i is the. In other words, a complex number is represented by the (a, b) over the field of real numbers. So the complex numbers form a two-dimensional real vector space, where addition is given by (a, b) + (c, d) = (a + c, b + d) and scalar multiplication is given by c(a, b) = (ca, cb), where all of a, b, c and d are real numbers. We use the symbol · to multiply two vectors together, which we use complex multiplication to define: (a, b) · (c, d) = (ac − bd, ad + bc).

The following statements are basic properties of the complex numbers. If x, y, z are complex numbers and a, b are real numbers, then


 * (x + y) · z = (x · z) + (y · z). In other words, multiplying a complex number by the sum of two other complex numbers, is the same as multiplying by each number in the sum, and then adding.
 * (ax) · (by) = (ab) (x · y). This shows that complex multiplication is compatible with the scalar multiplication by the real numbers.

This example fits into the following definition by taking the field K to be the real numbers, and the vector space A to be the complex numbers.

Definition
Let K be a field, and let A be a over K equipped with an additional  from A &times; A to A, denoted here by · (i.e. if x and y are any two elements of A, x · y is the product of x and y). Then A is an algebra over K if the following identities hold for all elements x, y, and z of A, and all elements (often called s) a and b of K:
 * Right : (x + y) · z = x · z + y · z
 * Left distributivity: z · (x + y) = z · x + z · y
 * Compatibility with scalars: (ax) · (by) = (ab) (x · y).

These three axioms are another way of saying that the binary operation is. An algebra over K is sometimes also called a K-algebra, and K is called the base field of A. The binary operation is often referred to as multiplication in A. The convention adopted in this article is that multiplication of elements of an algebra is not necessarily, although some authors use the term algebra to refer to an.

Notice that when a binary operation on a vector space is, as in the above example of the complex numbers, it is left distributive exactly when it is right distributive. But in general, for non-commutative operations (such as the next example of the quaternions), they are not equivalent, and therefore require separate axioms.

A motivating example: quaternions
The may be viewed as a one-dimensional vector space with a compatible multiplication, and hence a one-dimensional algebra over itself. Likewise, as we saw above, the complex numbers form a two-dimensional vector space over the field of real numbers, and hence form a two dimensional algebra over the reals. In both these examples, every has an, making them both. Although there are no division algebras in 3 dimensions, in 1843, the were defined and provided the now famous 4-dimensional example of an algebra over the real numbers, where one can not only multiply vectors, but also divide. Any quaternion may be written as (a, b, c, d) = a + bi + cj + dk. Unlike the complex numbers, the quaternions are an example of a algebra: for instance, (0,1,0,0) · (0,0,1,0) = (0,0,0,1) but (0,0,1,0) · (0,1,0,0) = (0,0,0,&minus;1).

The quaternions were soon followed by several other systems, which were the early examples of algebras over a field.

Another motivating example: the cross product
The previous examples are associative algebras. An example of a is a three dimensional vector space equipped with the. This is a simple example of a class of nonassociative algebras, which is widely used in and, the s.

Algebra homomorphisms
Given K-algebras A and B, a K-algebra is a K- f: A → B such that f(xy) = f(x) f(y) for all x, y in A. The space of all K-algebra homomorphisms between A and B is frequently written as
 * $$\mathbf{Hom}_{K\text{-alg}} (A,B).$$

A K-algebra is a  K-algebra homomorphism. For all practical purposes, isomorphic algebras differ only by notation.

Subalgebras and ideals
A subalgebra of an algebra over a field K is a 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 subset of elements that is closed under addition, multiplication, and scalar multiplication. In symbols, we say that a subset L of a K-algebra A is a subalgebra if for every x, y in L and c in K, we have that x · y, x + y, and cx are all in L.

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 K-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 L of a K-algebra A is a left ideal if for every x and y in L, z in A and c in K, we have the following three statements.
 * 1) x + y is in L (L is closed under addition),
 * 2) cx is in L (L is closed under scalar multiplication),
 * 3) z · x is in L (L is closed under left multiplication by arbitrary elements).

If (3) were replaced with x · z is in L, 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. Notice that conditions (1) and (2) together are equivalent to L being a linear subspace of A. It follows from condition (3) that every left or right ideal is a subalgebra.

It is important to notice that this definition is different from the definition of an, in that here we require the condition (2). Of course if the algebra is unital, then condition (3) implies condition (2).

Extension of scalars
If we have a F/K, which is to say a bigger field F that contains K, then there is a natural way to construct an algebra over F from any algebra over K. It is the same construction one uses to make a vector space over a bigger field, namely the tensor product $$ V_F:=V \otimes_K F $$. So if A is an algebra over K, then $$A_F$$ is an algebra over F.

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 or  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 or identity element I with Ix = x = xI for all x in the algebra.

Zero algebra
An algebra is called zero algebra if uv = 0 for all u, v in the algebra, not to be confused with the algebra with one element. It is inherently non-unital (except in the case of only one element), associative and commutative.

One may define a unital zero algebra by taking the of a field (or more generally a ring) K and a K-vector space (or module) V, and defining the product of every pair of elements of V to be zero. That is, if λ, μ ∈ K and u, v ∈ V, then (λ + u) (μ + v) = λμ + (λv + μu). If e1, ... ed is a basis of V, the unital zero algebra is the quotient of the polynomial ring K[E1, ..., En] by the generated by the EiEj for every pair (i, j).

An example of unital zero algebra is the algebra of s, the unital zero R-algebra built from a one dimensional real vector space.

These unital zero algebras may be more generally useful, as they allow to translate any general property of the algebras to properties of vector spaces or. For example, the theory of was introduced by  for  in a polynomial ring R = K[x1, ..., xn] over a field. The construction of the unital zero algebra over a free R-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.

Associative algebra
Examples of associative algebras include
 * the algebra of all n-by-n over a field (or commutative ring) K. Here the multiplication is ordinary.
 * s, where a serves as a basis of the vector space and algebra multiplication extends group multiplication.
 * the commutative algebra K[x] of all s over K (see ).
 * algebras of s, such as the R-algebra of all real-valued functions defined on the  [0,1], or the C-algebra of all s defined on some fixed open set in the . These are also commutative.
 * s are built on certain s.
 * algebras of s, for example on a . Here the algebra multiplication is given by the of operators. These algebras also carry a ; many of them are defined on an underlying, which turns them into s. If an involution is given as well, we obtain s and s. These are studied in.

Non-associative algebra
A non-associative algebra (or distributive algebra) over a field K is a K-vector space A equipped with a K- $$A \times A \rightarrow A$$. 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:
 * R3 with multiplication given by the
 * s
 * s
 * s
 * s
 * s
 * s

Algebras and rings
The definition of an associative K-algebra with unit is also frequently given in an alternative way. In this case, an algebra over a field K is a A together with a


 * $$\eta\colon K\to Z(A),$$

where Z(A) is the of A. Since η is a ring homomorphism, then one must have either that A is the, or that η is. This definition is equivalent to that above, with scalar multiplication
 * $$K\times A \to A$$

given by
 * $$(k,a) \mapsto \eta(k) a.$$

Given two such associative unital K-algebras A and B, a unital K-algebra homomorphism f: A → B is a ring homomorphism that commutes with the scalar multiplication defined by η, which one may write as
 * $$f(ka)=kf(a)$$

for all $$k\in K$$ and $$a \in A$$. In other words, the following diagram commutes:
 * $$\begin{matrix}

&& K &&  \\ & \eta_A \swarrow & \, & \eta_B \searrow & \\ A && \begin{matrix} f \\ \longrightarrow \end{matrix}  && B \end{matrix}$$

Structure coefficients
For algebras over a field, the bilinear multiplication from A &times; A to A is completely determined by the multiplication of elements of A. Conversely, once a basis for A has been chosen, the products of basis elements can be set arbitrarily, and then extended in a unique way to a bilinear operator on A, i.e., so the resulting multiplication satisfies the algebra laws.

Thus, given the field K, any finite-dimensional algebra can be specified  by giving its  (say n), and specifying n3 structure coefficients ci,j,k, which are. These structure coefficients determine the multiplication in A via the following rule:
 * $$\mathbf{e}_{i} \mathbf{e}_{j} = \sum_{k=1}^n c_{i,j,k} \mathbf{e}_{k}$$

where e1,...,en form a basis of A.

Note however that several different sets of structure coefficients can give rise to isomorphic algebras.

In, the structure coefficients are generally written with upper and lower indices, so as to distinguish their transformation properties under coordinate transformations. Specifically, lower indices are indices, and transform via s, while upper indices are, transforming under s. Thus, the structure coefficients are often written ci,jk, and their defining rule is written using the  as
 * eiej = ci,jkek.

If you apply this to vectors written in, then this becomes
 * (xy)k = ci,jkxiyj.

If K is only a commutative ring and not a field, then the same process works if A is a over K. If it isn't, then the multiplication is still completely determined by its action on a set that spans A; however, the structure constants can't be specified arbitrarily in this case, and knowing only the structure constants does not specify the algebra up to isomorphism.

Classification of low-dimensional algebras
Two-dimensional, three-dimensional and four-dimensional unital associative algebras over the field of complex numbers were completely classified up to isomorphism by.

There exist two two-dimensional algebras. Each algebra consists of linear combinations (with complex coefficients) of two basis elements, 1 (the identity element) and a. According to the definition of an identity element,
 * $$\textstyle 1 \cdot 1 = 1 \,, \quad 1 \cdot a = a \, , \quad a \cdot 1 = a \, . $$

It remains to specify
 * $$\textstyle a a = 1 $$  for the first algebra,
 * $$\textstyle a a = 0 $$  for the second algebra.

There exist five three-dimensional algebras. Each algebra consists of linear combinations of three basis elements, 1 (the identity element), a and b. Taking into account the definition of an identity element, it is sufficient to specify
 * $$\textstyle a a = a \,, \quad b b = b \, , \quad a b = b a = 0 $$  for the first algebra,
 * $$\textstyle a a = a \,, \quad b b = 0 \, , \quad a b = b a = 0 $$  for the second algebra,
 * $$\textstyle a a = b \,, \quad b b = 0 \, , \quad a b = b a = 0 $$  for the third algebra,
 * $$\textstyle a a = 1 \,, \quad b b = 0 \, , \quad a b = - b a = b $$  for the fourth algebra,
 * $$\textstyle a a = 0 \,, \quad b b = 0 \, , \quad a b = b a = 0 $$  for the fifth algebra.

The fourth algebra is non-commutative, and the others are commutative.

Generalization: algebra over a ring
In some areas of mathematics, such as, it is common to consider the more general concept of an algebra over a ring, where a commutative unital ring R replaces the field K. The only part of the definition that changes is that A is assumed to be an (instead of a vector space over K).

Associative algebras over rings
A A is always an associative algebra over its, and over the s. A classical example of an algebra over its center, is the , which is isomorphic to $$\mathbb{H} \times \mathbb{H}$$, the direct product of two. The center of that ring is $$\mathbb{R} \times \mathbb{R}$$, and hence it has the structure of an algebra over its center, which is not a field. Note that the split-biquaternion algebra is also naturally an 8-dimensional $$\mathbb{R}$$-algebra.

In commutative algebra, if A is a, then any unital ring homomorphism $$R \to A$$ defines an R-module structure on A, and this is what is known as the R-algebra structure. So a ring comes with a natural $$\mathbb{Z}$$-module structure, since one can take the unique homomorphism $$\mathbb{Z} \to A$$. On the other hand, not all rings can be given the structure of an algebra over a field (for example the integers). See the for an attempt to give to every ring a structure that behaves like an algebra over a field.