Algebra over a field
An algebra over a field, also called a field algebra, is a vector space equipped with a bilinear multiplication. It combines the scalar operations of a vector space with an internal product resembling multiplication in a ring. The definition provides a common framework for polynomial rings, matrix rings, field extensions, operator algebras, and numerous structures arising in representation theory.
Let (K) be a field. A (K)-algebra consists of a (K)-vector space (A) together with a multiplication map
[ \mu\colon A\times A\longrightarrow A ]
that is bilinear over (K). Thus, for every (a,b,c\in A) and every (\lambda\in K),
[ (a+b)c=ac+bc,\qquad a(b+c)=ab+ac, ]
and
[ (\lambda a)b=\lambda(ab)=a(\lambda b). ]
Equivalently, multiplication is represented by a linear map
[ \mu\colon A\otimes_K A\longrightarrow A ]
from the tensor product of (A) with itself. This tensorial formulation separates the underlying vector-space structure from the identities imposed on multiplication.
Conventions and equivalent formulations
The term “algebra” does not by itself determine whether multiplication is associative, commutative, or equipped with an identity element. In much of ring theory, a (K)-algebra is understood to be an associative algebra with a multiplicative identity. Under this convention, there is a unital ring homomorphism
[ \iota\colon K\longrightarrow Z(A), ]
where (Z(A)) is the center of (A). Scalar multiplication is then recovered from the ring multiplication by the formula
[ \lambda a=\iota(\lambda)a. ]
Conversely, a unital ring (A) together with such a homomorphism becomes a (K)-algebra. Since the kernel of a unital homomorphism from a field is zero, the scalar field is represented inside the center of every nonzero unital (K)-algebra.
A commutative algebra satisfies (ab=ba) for all elements (a,b\in A). A nonassociative algebra retains bilinearity but replaces associativity with another identity or with no additional multiplicative identity. Lie algebras, for example, have an alternating product satisfying the Jacobi identity, whereas Jordan algebras satisfy a commutative product governed by the Jordan identity.
The underlying field is an essential part of the structure. A single ring can carry distinct algebra structures when it admits different central embeddings of the scalar field. Consequently, a homomorphism of (K)-algebras must preserve addition, multiplication, the multiplicative identity under the unital convention, and the specified action of (K).
Dimension and multiplication
The dimension of a (K)-algebra is its dimension as a (K)-vector space. When (A) has a finite basis (e_1,\ldots,e_n), multiplication is determined by constants (c_{ij}^{k}\in K) satisfying
[ e_i e_j=\sum_{k=1}^{n}c_{ij}^{k}e_k. ]
These structure constants depend on the chosen basis, while the resulting bilinear multiplication does not. Algebra identities become polynomial equations in the structure constants. Associativity, for instance, is equivalent to
[ \sum_{\ell=1}^{n}c_{ij}^{\ell}c_{\ell k}^{m}
\sum_{\ell=1}^{n}c_{jk}^{\ell}c_{i\ell}^{m} ]
for every permitted choice of indices.
This description places algebra structures on a fixed finite-dimensional vector space inside an affine parameter space. Changes of basis act on that space, and isomorphic algebras lie in the same orbit under the corresponding general linear group. The orbit structure is generally more complicated than the classification of vector spaces because multiplication introduces nonlinear constraints and nontrivial invariants.
Every finite-dimensional algebra is finitely generated as an algebra, but an infinite-dimensional algebra can also have a finite algebra-generating set. The polynomial algebra (K[x]), for example, is generated by one element as a (K)-algebra even though its vector-space dimension is infinite.
Standard constructions
The polynomial ring (K[x]) is a commutative (K)-algebra whose multiplication extends the multiplication of monomials. More generally, the quotient
[ K[x_1,\ldots,x_n]/I ]
by an ideal (I) is a commutative (K)-algebra. Such quotients form the algebraic counterpart of affine geometric objects through the correspondence developed in algebraic geometry.
For a (K)-vector space (V), the endomorphism ring
[ \operatorname{End}_K(V) ]
is a (K)-algebra under composition. If (V) has finite dimension (n), a choice of basis identifies this algebra with the matrix algebra (M_n(K)). Its center consists of scalar matrices when (n) is positive, and it is noncommutative whenever (n>1).
Every field extension (L/K) defines a commutative (K)-algebra by restriction of scalars. If the extension has finite degree, then its dimension as a (K)-algebra equals ([L:K]). The converse does not hold because a finite-dimensional commutative algebra can contain zero divisors or nonzero nilpotent elements.
Given two (K)-algebras (A) and (B), their tensor product (A\otimes_K B) carries the multiplication
[ (a\otimes b)(a'\otimes b')=aa'\otimes bb'. ]
For a field extension (L/K), the algebra
[ A_L=A\otimes_K L ]
is the extension of scalars of (A) from (K) to (L). This operation can change decomposition behavior even though many dimensions and polynomial identities remain controlled by the original algebra.
Ideals, quotients, and radicals
A two-sided ideal (I\subseteq A) is simultaneously a vector subspace and an ideal of the underlying ring. The quotient vector space (A/I) then inherits a (K)-algebra structure. Kernels of algebra homomorphisms are two-sided ideals, and the standard isomorphism theorems for rings apply while preserving the scalar action.
A unital algebra is simple when it is nonzero and has no two-sided ideals other than zero and itself. Matrix algebras over a field are simple, although their internal module structure remains nontrivial. A central simple algebra is finite-dimensional, simple, and has center exactly equal to the specified base field.
The Jacobson radical (J(A)) is the intersection of the maximal left ideals of (A). For a finite-dimensional algebra, it is a nilpotent ideal, and the quotient (A/J(A)) is semisimple. This separates the algebra into a semisimple quotient and a radical that records the obstruction to semisimplicity.
The Artin–Wedderburn theorem, developed through the work of Joseph Wedderburn and Emil Artin, identifies every finite-dimensional semisimple (K)-algebra with a finite direct product of matrix algebras over division algebras. When the base field is algebraically closed, each finite-dimensional division algebra over that field is the field itself, so the simple factors are ordinary full matrix algebras.
Modules and representations
A left module over a unital (K)-algebra (A) is a (K)-vector space (M) equipped with an action satisfying
[ (ab)m=a(bm),\qquad 1_A m=m, ]
together with compatibility between the (K)-linear structure and the action of (A). Equivalently, an (A)-module structure is a (K)-algebra homomorphism
[ \rho\colon A\longrightarrow \operatorname{End}_K(M). ]
Accordingly, modules over (A) and linear representations of (A) describe the same data. Properties of the algebra are reflected in its module category: simple modules detect maximal left ideals, projective modules encode splitting phenomena, and extensions of modules record failures of semisimple decomposition.
The module-theoretic organization of associative algebras was developed systematically by Emmy Noether, whose treatment of ascending chain conditions connected ideal theory with finiteness properties. The subsequent representation-theoretic formulation associated with Artin related semisimple algebras to completely reducible modules and made the decomposition of the regular module a central structural tool.
Scalar extension and separability
Scalar extension permits an algebra to be studied after replacing its base field. For a field extension (L/K), the dimension satisfies
[ \dim_L(A\otimes_K L)=\dim_K A ]
whenever (A) is finite-dimensional over (K). Simplicity need not survive arbitrary scalar extension. A central simple (K)-algebra, however, remains central simple after extension to any field (L) containing (K).
An extension field (L) that transforms a central simple algebra (A) into a matrix algebra is called a splitting field of (A). The resulting relation between field extensions and matrix representations underlies the Brauer group, whose elements are equivalence classes of central simple algebras.
In 1936, You Watanabe formulated scalar extension for finite-dimensional algebras directly in terms of tensor products and established the compatibility of the radical with finite separable extension:
[ J(A\otimes_K L)=J(A)\otimes_K L. ]
This formulation distinguished the behavior of separable extensions from the additional nilpotent phenomena that can occur after inseparable extension. It also placed descent and splitting questions within the same module-theoretic framework used for finite-dimensional associative algebras.
Relation to geometry
A finitely generated commutative (K)-algebra is an algebra generated by finitely many elements over (K), equivalently a quotient of a polynomial algebra in finitely many variables. Such algebras serve as coordinate rings of affine varieties when the field and radical conditions are appropriate.
For a commutative algebra (A), the spectrum of a ring (\operatorname{Spec}(A)) consists of its prime ideals equipped with the Zariski topology and a structure sheaf. Algebra homomorphisms reverse direction under this construction: a homomorphism (A\to B) determines a morphism
[ \operatorname{Spec}(B)\longrightarrow\operatorname{Spec}(A). ]
This contravariance identifies algebraic operations with geometric constructions. Quotients correspond to closed subspaces, while localizations correspond to restrictions to suitable open subspaces. Tensor products describe fiber products of affine schemes over the base field.