In mathematics, an algebra over a field is a vector space equipped with a bilinear product. An algebra such that the product is associative and has an identity is therefore a ring that is also a vector space, and thus equipped with a field of scalars. Such an algebra is called here a unital associative algebra for clarity, because there are also nonassociative algebras.
In other words, an algebra over a field is a set together with operations of multiplication, addition, and scalar multiplication by elements of the underlying field, that satisfy the axioms implied by "vector space" and "bilinear".^{[1]}
One may generalize this notion by replacing the field of scalars by a commutative ring, and thus defining an algebra over a ring.
Because of the ubiquity of associative algebras, and because many textbooks teach more associative algebra than nonassociative algebra, it is common for authors to use the term algebra to mean associative algebra. However, this does not diminish the importance of nonassociative algebras, and there are texts that give both structures and names equal priority.
Definition and motivation
First example: The complex numbers
Any complex number may be written a + bi, where a and b are real numbers and i is the imaginary unit. In other words, a complex number is represented by the vector (a, b) over the field of real numbers. So the complex numbers form a twodimensional 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. Let x, y, z be complex numbers, and let a, b be real numbers.


(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 vector space over K equipped with an additional binary operation from A × 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 any three elements x, y, and z of A, and all elements ("scalars") a and b of K:

Right distributivity: (x + y) · z = x · z + y · z

Left distributivity: x · (y + z) = x · y + x · z

Compatibility with scalars: (ax) · (by) = (ab) (x · y).
These three axioms are another way of saying that the binary operation is bilinear. An algebra over K is sometimes also called a Kalgebra, 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 associative, although some authors use the term algebra to refer to an associative algebra.
Notice that when a binary operation on a vector space is commutative, as in the above example of the complex numbers, it is left distributive exactly when it is right distributive. But in general, for noncommutative operations (such as the next example of the quaternions), they are not equivalent, and therefore require separate axioms.
A motivating example: quaternions
The real numbers may be viewed as a onedimensional vector space with a compatible multiplication, and hence a onedimensional algebra over itself. Likewise, as we saw above, the complex numbers form a twodimensional vector space over the field of real numbers, and hence form a two dimensional algebra over the reals. In both these examples, every nonzero vector has an inverse, making them both division algebras. It is natural to ask whether one can similarly define a multiplication on a threedimensional real vector space such that every nonzero element has an inverse. The answer is no (see normed division algebras).
Although there are no division algebras in 3 dimensions, in 1843, the quaternions were defined and provided the now famous 4dimensional 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 noncommutative 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,−1).
The quaternions were soon followed by several other hypercomplex number systems, which were the early examples of algebras over a field.
Another motivating example: the cross product
Previous examples are associative algebras. An example of a nonassociative algebra is a three dimensional vector space equipped with the cross product. This is a simple example of a class of nonassociative algebras, which is widely used in mathematics and physics, the Lie algebras.
Basic concepts
Algebra homomorphisms
Given Kalgebras A and B, a Kalgebra homomorphism is a Klinear map f: A → B such that f(xy) = f(x) f(y) for all x,y in A. The space of all Kalgebra homomorphisms between A and B is frequently written as

\mathbf{Hom}_{K\text{alg}} (A,B).
A Kalgebra isomorphism is a bijective Kalgebra morphism. For all practical purposes, isomorphic algebras differ only by notation.
Subalgebras and ideals
A subalgebra of an algebra over a field K 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 subset of elements that is closed under addition, multiplication, and scalar multiplication. In symbols, we say that a subset L of a Kalgebra 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 twodimensional algebra over the real numbers, the onedimensional real line is a subalgebra.
A left ideal of a Kalgebra 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 Kalgebra 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 twosided 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 twosided 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 ideal of a ring, 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 field extension 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 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 algebras
An algebra is unital or unitary if it has a unit or identity element I with Ix = x = xI for all x in the algebra.
Zero algebras
An algebra is called zero algebra if uv = 0 for all u, v in the algebra,^{[2]} not to be confused with the algebra with one element. It is inherently nonunital (except in the case of only one element), associative and commutative.
One may define a unital zero algebra by taking the direct sum of modules of a field (or more generally a ring) k and a k vector space (or module) V, and defining the product of two 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 e_{1}, ... e_{d} is a basis of V, the unital zero algebra is the quotient of the polynomial ring k[E_{1}, ..., E_{n}] by the ideal generated by the E_{i}E_{j} for every pair (i,j).
An example of unital zero algebra is the algebra of dual numbers, which is the unital zero Ralgebra which is 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 modules. For example, the theory of Gröbner bases was introduced by Bruno Buchberger for ideals in a polynomial ring R = k[x_{1}, ..., x_{n}] over a field. The construction of the unital zero algebra over a free Rmodule allows to extend directly this theory as a Gröbner basis theory for sub modules 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 algebras
Nonassociative algebras
A nonassociative algebra^{[3]} (or distributive algebra) over a field K is a Kvector space A equipped with a Kbilinear map A \times A \rightarrow A. The usage of "nonassociative" 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" just as "noncommutative" means "not necessarily commutative".
Examples detailed in the main article include:
Algebras and rings
The definition of an associative Kalgebra with unit is also frequently given in an alternative way. In this case, an algebra over a field K is a ring A together with a ring homomorphism

\eta\colon K\to Z(A),
where Z(A) is the center (algebra) of A. Since η is a ring morphism, then one must have either that A is the trivial ring, or that η is injective. 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 Kalgebras A and B, a unital Kalgebra morphism f: A → B is a ring morphism 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 × A to A is completely determined by the multiplication of basis 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 finitedimensional algebra can be specified up to isomorphism by giving its dimension (say n), and specifying n^{3} structure coefficients c_{i,j,k}, which are scalars. 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 e_{1},...,e_{n} form a basis of A.
Note however that several different sets of structure coefficients can give rise to isomorphic algebras.
When the algebra can be endowed with a metric, then the structure coefficients are written with upper and lower indices, so as to distinguish their transformation properties under coordinate transformations. Specifically, lower indices are covariant indices, and transform via pullbacks, while upper indices are contravariant, transforming under pushforwards. Thus, in mathematical physics, the structure coefficients are often written c_{i,j}^{k}, and their defining rule is written using the Einstein notation as

e_{i}e_{j} = c_{i,j}^{k}e_{k}.
If you apply this to vectors written in index notation, then this becomes

(xy)^{k} = c_{i,j}^{k}x^{i}y^{j}.
If K is only a commutative ring and not a field, then the same process works if A is a free module 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 lowdimensional algebras
Twodimensional, threedimensional and fourdimensional unital associative algebras over the field of complex numbers were completely classified up to isomorphism by Eduard Study.^{[4]}
There exist two twodimensional 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 threedimensional 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 noncommutative, others are commutative.
See also
Notes

^ See also Hazewinkel et al. (2004). Algebras, rings and modules 1. p. 3.

^ João B. Prolla, Approximation of vector valued functions, Elsevier, 1977, p. 65

^ Richard D. Schafer, An Introduction to Nonassociative Algebras (1996) ISBN 0486688135 Gutenberg eText

^ Study, E. (1890), "Über Systeme complexer Zahlen und ihre Anwendungen in der Theorie der Transformationsgruppen", Monatshefte fũr Mathematik 1 (1): 283–354,
References

Michiel Hazewinkel, Nadiya Gubareni, Nadezhda Mikhaĭlovna Gubareni, Vladimir V. Kirichenko. Algebras, rings and modules. Volume 1. 2004. Springer, 2004. ISBN 1402026900
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