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体上的代数（algebra over a field）或体代数，一般可简称为代数，是在向量空间的基础上定义了一个双线性的乘法运算而构成的代数结构。根据此乘法是否具有结合律，可以进一步地分成结合代数以及非结合代数两类。如果乘法单位元包含在此代数里，则称为单位代数。

若没有特别指明，通常假设此代数为结合代数。而在一些代数几何的讨论框架下，会假设此代数是、单位结合且交换。在更一般的情况下，会讨论将向量空间换成环所形成的代数，称为环上的代数（algebra over a ring）。

需要注意的是这里的双线性乘法运算跟向量空间上的双线性形式是不一样的。具体而言，双线性乘法运算是一个在向量空间里的向量，而双线性形式所给出的是在体K上的纯量。

定义

例子

代数	向量空间	双线性乘法	结合律	交换律
复数	$\mathbb {R} ^{2}$	复数里的乘法 $\left(a+ib\right)\cdot \left(c+id\right)$	有	有
三围向量的外积	$\mathbb {R} ^{3}$	外积 ${\vec {a}}\times {\vec {b}}$	无	无
四元数s	$\mathbb {R} ^{4}$	Hamilton product $(a+{\vec {v}})(b+{\vec {w}})$	有	无
多项式	$\mathbb {R} [X]$	多项式乘法	有	有
方块矩阵	$\mathbb {R} ^{n\times n}$	矩阵乘法	有	无

定义

令K为一个体, A为 K上的向量空间，且有二元乘法，记为 $\cdot :A\times A\to A\ ;\ (x,y)\mapsto A$ 则A 是一个K-代数（K-algebra）如果满足以下几点：

对于所有 A中的向量 x, y, z ，所有K中的纯量 a,b，有

• 右分配律：(x + y) · z = x · z + y · z
• 左分配律：z · (x + y) = z · x + z · y
• 与纯量相容：(ax) · (by) = (ab) (x · y)

此时的K称为A的基体（base field）。此外，当A有交换律时，左分配律等价于右分配律。

基本概念

代数上的同态

给定 K-代数 A 以及 B,两个 K-algebras 上的同态（ K-algebra homomorphism）是一个K-线性映射 f: A → B 使得对于所有 A 中的 x, y ，都有 f(xy) = f(x) f(y)。若 A 、B 都是单位代数，则满足f(1_A) = 1_B 的同态称为单位同态（unital homomorphism）。所有K-algebras 上的同态所构成的空间通常写成：

$\mathbf {Hom} _{K{\text{-alg}}}(A,B).$

K-algebras 上的同构（ K-algebra isomorphism）是双射的K-algebras 上的同态。

子代数

一个K-代数的子代数是一个线性子空间，且具有乘法封闭性（任何两个元素的乘积仍然在这个子空间内）。换句话说，代数的一个子代数是一个在加法、乘法和纯量乘法下闭合的非空子集。

形式上，给定 K-代数 A ，$L\subseteq A$ 为一子集，且满足以下条件

$\forall x,y\in L\ ,\ \forall c\in K\quad x\cdot y,\ y\cdot x,\ cx\in L$，则称L 是一个子代数。一个例子是考虑 $\mathbb {C} =\mathbb {R} ^{2}$，则$\mathbb {R}$形成一个子代数。

理想

一个 K 代数的左理想是一个线性子空间，具有这样的性质：子空间中的任何元素与代数中的任何元素在左侧相乘后仍在这个子空间内。用符号来表示，我们说 K-代数 A 的一个子集 L 是一个左理想，如果对于 L 中的所有元素 x 和 y，代数 A 中的元素 z 和 K 中的纯量 c，满足以下三点：

1. x+y 在 L 中（L 在加法下闭合），
2. cx 在 L 中（L 在纯量乘法下闭合），
3. z⋅x 在 L 中（L 在左乘任意元素下闭合）。

若将（3）替换为 x⋅z 在 L 中，则这将定义右理想。双边理想是同时是左理想和右理想的子集。理想这个术语通常指双边理想，而当代数是交换代数时，左理想等价于右理想等价于双边理想。条件（1）和（2）一起等价于 L 是 A 的线性子空间。根据条件（3），每个左理想或右理想都是子代数。这个定义与环的理想的定义不同，因为在这里我们要求条件（2），包含了额外的纯量。当然，如果代数是带单位元的，则条件（3）蕴含条件（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 algebra

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 algebra

An algebra is called a 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 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 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, ... e_d is a basis of V, the unital zero algebra is the quotient of the polynomial ring K[E₁, ..., E_n] by the ideal generated by the E_iE_j for every pair (i, j).

An example of unital zero algebra is the algebra of dual numbers, 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 modules. For example, the theory of Gröbner bases was introduced by Bruno Buchberger for ideals in a polynomial ring R = K[x₁, ..., x_n] 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

主条目：Associative algebra

Examples of associative algebras include

• the algebra of all n-by-n matrices over a field (or commutative ring) K. 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 K[x] of all polynomials over K (see polynomial ring).
• algebras of functions, such as the R-algebra of all real-valued continuous functions defined on the interval [0,1], or the C-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

主条目：Non-associative algebra

A non-associative algebra (or distributive algebra) over a field K is a K-vector space A equipped with a K-bilinear map $A\times A\rightarrow A$. 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:

• Euclidean space R³ 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 K-algebra 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 of A. Since η is a ring homomorphism, then one must have either that A is the zero 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 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

主条目：Structure constants

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 finite-dimensional algebra can be specified up to isomorphism by giving its dimension (say n), and specifying n³ 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 e₁,...,e_n form a basis of A.

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

In mathematical physics, 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 covariant indices, and transform via pullbacks, while upper indices are contravariant, transforming under pushforwards. Thus, the structure coefficients are often written c_i,j^k, and their defining rule is written using the Einstein notation as

e_ie_j = c_i,j^ke_k.

If you apply this to vectors written in index notation, then this becomes

(xy)^k = c_i,j^kxⁱy^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 low-dimensional unital associative algebras over the complex numbers

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

There exist two such 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

$\textstyle aa=1$   for the first algebra,

$\textstyle aa=0$   for the second algebra.

There exist five such 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

$\textstyle aa=a\,,\quad bb=b\,,\quad ab=ba=0$   for the first algebra,

$\textstyle aa=a\,,\quad bb=0\,,\quad ab=ba=0$   for the second algebra,

$\textstyle aa=b\,,\quad bb=0\,,\quad ab=ba=0$   for the third algebra,

$\textstyle aa=1\,,\quad bb=0\,,\quad ab=-ba=b$   for the fourth algebra,

$\textstyle aa=0\,,\quad bb=0\,,\quad ab=ba=0$   for the fifth algebra.

The fourth of these algebras is non-commutative, and the others are commutative.

Generalization: algebra over a ring

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

Associative algebras over rings

主条目：Associative algebra

A ring A is always an associative algebra over its center, and over the integers. A classical example of an algebra over its center is the split-biquaternion algebra, which is isomorphic to $\mathbb {H} \times \mathbb {H}$, the direct product of two quaternion algebras. 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 commutative ring, 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 Field with one element for a description of an attempt to give to every ring a structure that behaves like an algebra over a field.

See also

• Algebra over an operad
• Alternative algebra
• Clifford algebra
• Differential algebra
• Free algebra
• Geometric algebra
• Max-plus algebra
• Mutation (algebra)
• Operator algebra
• Zariski's lemma

References

• Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, Vladimir V. Algebras, rings and modules 1. Springer. 2004. ISBN 1-4020-2690-0.