In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an associative algebra ''A'' over a
commutative ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
(often a
field) ''K'' is a
ring ''A'' together with a
ring homomorphism
In mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function that preserves addition, multiplication and multiplicative identity ...
from ''K'' into the
center of ''A''. This is thus an
algebraic structure
In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
with an addition, a multiplication, and a
scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector ...
(the multiplication by the image of the ring homomorphism of an element of ''K''). The addition and multiplication operations together give ''A'' the structure of a
ring; the addition and scalar multiplication operations together give ''A'' the structure of a
module or
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
over ''K''. In this article we will also use the term
''K''-algebra to mean an associative algebra over ''K''. A standard first example of a ''K''-algebra is a ring of
square matrices over a commutative ring ''K'', with the usual
matrix multiplication
In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...
.
A commutative algebra is an associative algebra for which the multiplication is
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
, or, equivalently, an associative algebra that is also a
commutative ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
.
In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification. In some areas of mathematics this assumption is not made, and we will call such structures
non-unital associative algebras. We will also assume that all rings are unital, and all ring homomorphisms are unital.
Every ring is an associative algebra over its center and over the integers.
Definition
Let ''R'' be a
commutative ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
(so ''R'' could be a field). An associative ''R''-algebra ''A'' (or more simply, an ''R''-algebra ''A'') is a
ring ''A''
that is also an
''R''-module in such a way that the two additions (the ring addition and the module addition) are the same operation, and
scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector ...
satisfies
:
for all ''r'' in ''R'' and ''x'', ''y'' in the algebra. (This definition implies that the algebra, being a ring, is
unital, since rings are supposed to have a
multiplicative identity
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
.)
Equivalently, an associative algebra ''A'' is a ring together with a
ring homomorphism
In mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function that preserves addition, multiplication and multiplicative identity ...
from ''R'' to the
center of ''A''. If ''f'' is such a homomorphism, the scalar multiplication is (here the multiplication is the ring multiplication); if the scalar multiplication is given, the ring homomorphism is given by . (See also ' below).
Every ring is an associative Z-algebra, where Z denotes the ring of the
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s.
A is an associative algebra that is also a
commutative ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
.
As a monoid object in the category of modules
The definition is equivalent to saying that a unital associative ''R''-algebra is a
monoid object in
''R''-Mod (the
monoidal category
In mathematics, a monoidal category (or tensor category) is a category (mathematics), category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an Object (cate ...
of ''R''-modules). By definition, a ring is a monoid object in the
category of abelian groups
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab.
Properties
The zero object o ...
; thus, the notion of an associative algebra is obtained by replacing the category of abelian groups with the
category of modules
In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ...
.
Pushing this idea further, some authors have introduced a "generalized ring" as a monoid object in some other category that behaves like the category of modules. Indeed, this reinterpretation allows one to avoid making an explicit reference to elements of an algebra ''A''. For example, the associativity can be expressed as follows. By the universal property of a
tensor product of modules
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor produ ...
, the multiplication (the ''R''-bilinear map) corresponds to a unique ''R''-linear map
:
.
The associativity then refers to the identity:
:
From ring homomorphisms
An associative algebra amounts to a
ring homomorphism
In mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function that preserves addition, multiplication and multiplicative identity ...
whose image lies in the
center. Indeed, starting with a ring ''A'' and a ring homomorphism whose image lies in the
center of ''A'', we can make ''A'' an ''R''-algebra by defining
:
for all and . If ''A'' is an ''R''-algebra, taking , the same formula in turn defines a ring homomorphism whose image lies in the center.
If a ring is commutative then it equals its center, so that a commutative ''R''-algebra can be defined simply as a commutative ring ''A'' together with a commutative ring homomorphism .
The ring homomorphism ''η'' appearing in the above is often called a
structure map. In the commutative case, one can consider the category whose objects are ring homomorphisms for a fixed ''R'', i.e., commutative ''R''-algebras, and whose morphisms are ring homomorphisms that are under ''R''; i.e., is (i.e., the
coslice category of the category of commutative rings under ''R''.) The
prime spectrum functor Spec then determines an
anti-equivalence of this category to the category of
affine scheme
In commutative algebra, the prime spectrum (or simply the spectrum) of a commutative ring R is the set of all prime ideals of R, and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with ...
s over Spec ''R''.
How to weaken the commutativity assumption is a subject matter of
noncommutative algebraic geometry and, more recently, of
derived algebraic geometry. See also: ''
Generic matrix ring''.
Algebra homomorphisms
A
homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
between two ''R''-algebras is an
''R''-linear ring homomorphism
In mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function that preserves addition, multiplication and multiplicative identity ...
. Explicitly, is an associative algebra homomorphism if
:
The class of all ''R''-algebras together with algebra homomorphisms between them form a
category
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* Category ( ...
, sometimes denoted ''R''-Alg.
The
subcategory of commutative ''R''-algebras can be characterized as the
coslice category ''R''/CRing where CRing is the
category of commutative rings.
Examples
The most basic example is a ring itself; it is an algebra over its
center or any subring lying in the center. In particular, any commutative ring is an algebra over any of its subrings. Other examples abound both from algebra and other fields of mathematics.
Algebra
* Any ring ''A'' can be considered as a Z-algebra. The unique ring homomorphism from Z to ''A'' is determined by the fact that it must send 1 to the identity in ''A''. Therefore, rings and Z-algebras are equivalent concepts, in the same way that
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
s and Z-modules are equivalent.
* Any ring of
characteristic ''n'' is a (Z/''n''Z)-algebra in the same way.
* Given an ''R''-module ''M'', the
endomorphism ring
In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in ...
of ''M'', denoted End
''R''(''M'') is an ''R''-algebra by defining .
* Any ring of
matrices with coefficients in a commutative ring ''R'' forms an ''R''-algebra under matrix addition and multiplication. This coincides with the previous example when ''M'' is a finitely-generated,
free ''R''-module.
** In particular, the square ''n''-by-''n''
matrices with entries from the field ''K'' form an associative algebra over ''K''.
* The
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s form a 2-dimensional commutative algebra over the
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s.
* The
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s form a 4-dimensional associative algebra over the reals (but not an algebra over the complex numbers, since the complex numbers are not in the center of the quaternions).
* Every
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...
is a commutative ''R''-algebra. In fact, this is the free commutative ''R''-algebra on the set .
* The
free ''R''-algebra on a set ''E'' is an algebra of "polynomials" with coefficients in ''R'' and noncommuting indeterminates taken from the set ''E''.
* The
tensor algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra over a field, algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', ...
of an ''R''-module is naturally an associative ''R''-algebra. The same is true for quotients such as the
exterior and
symmetric algebras. Categorically speaking, the
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
that maps an ''R''-module to its tensor algebra is
left adjoint
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are k ...
to the functor that sends an ''R''-algebra to its underlying ''R''-module (forgetting the multiplicative structure).
* Given a module ''M'' over a commutative ring ''R'', the direct sum of modules has a structure of an ''R''-algebra by thinking ''M'' consists of infinitesimal elements; i.e., the multiplication is given as . The notion is sometimes called the
algebra of dual numbers.
* A
quasi-free algebra, introduced by Cuntz and Quillen, is a sort of generalization of a free algebra and a semisimple algebra over an algebraically closed field.
Representation theory
* The
universal enveloping algebra of a Lie algebra is an associative algebra that can be used to study the given Lie algebra.
* If ''G'' is a group and ''R'' is a commutative ring, the set of all functions from ''G'' to ''R'' with finite support form an ''R''-algebra with the convolution as multiplication. It is called the
group algebra of ''G''. The construction is the starting point for the application to the study of (discrete) groups.
* If ''G'' is an
algebraic group (e.g., semisimple
complex Lie group), then the
coordinate ring
In algebraic geometry, an affine variety or affine algebraic variety is a certain kind of algebraic variety that can be described as a subset of an affine space.
More formally, an affine algebraic set is the set of the common zeros over an algeb ...
of ''G'' is the
Hopf algebra ''A'' corresponding to ''G''. Many structures of ''G'' translate to those of ''A''.
* A
quiver algebra (or a path algebra) of a directed graph is the free associative algebra over a field generated by the paths in the graph.
Analysis
* Given any
Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
''X'', the
continuous linear operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
s form an associative algebra (using composition of operators as multiplication); this is a
Banach algebra.
* Given any
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
''X'', the continuous real- or complex-valued functions on ''X'' form a real or complex associative algebra; here the functions are added and multiplied pointwise.
* The set of
semimartingales defined on the
filtered probability space forms a ring under
stochastic integration.
* The
Weyl algebra
* An
Azumaya algebra
Geometry and combinatorics
* The
Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure of a distinguished subspace. As -algebras, they generalize the real number ...
s, which are useful in
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
and
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
.
*
Incidence algebra
In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered set
and commutative ring with unity. Subalgebra#Subalgebras_for_algebras_over_a_ring_or_field, Subalgebras c ...
s of
locally finite partially ordered set
In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
s are associative algebras considered in
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
.
* The
partition algebra and its subalgebras, including the
Brauer algebra and the
Temperley-Lieb algebra.
* A
differential graded algebra
In mathematics – particularly in homological algebra, algebraic topology, and algebraic geometry – a differential graded algebra (or DGA, or DG algebra) is an algebraic structure often used to capture information about a topological or geo ...
is an associative algebra together with a grading and a differential. For example, the
de Rham algebra , where
consists of differential ''p''-forms on a manifold ''M'', is a differential graded algebra.
Mathematical physics
* A
Poisson algebra is a commutative associative algebra over a field together with a structure of a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
so that the Lie bracket satisfies the Leibniz rule; i.e., .
* Given a Poisson algebra
, consider the vector space