In mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center of A. This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication (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 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.
A commutative algebra is an associative algebra for which the multiplication is commutative, or, equivalently, an associative algebra that is also a commutative ring.
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 (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 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.)
Equivalently, an associative algebra A is a ring together with a ring homomorphism 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 integers.
A is an associative algebra that is also a commutative ring.
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 of R-modules). By definition, a ring is a monoid object in the category of abelian groups; thus, the notion of an associative algebra is obtained by replacing the category of abelian groups with the category of modules.
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, 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 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 schemes 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 between two R-algebras is an R-linear ring homomorphism. Explicitly, is an associative algebra homomorphism if
The class of all R-algebras together with algebra homomorphisms between them form a 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 groups and Z-modules are equivalent. Any ring of characteristic n is a (Z/nZ)-algebra in the same way. Given an R-module M, the endomorphism ring of M, denoted EndR(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 numbers form a 2-dimensional commutative algebra over the real numbers. The quaternions 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 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 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 that maps an R-module to its tensor algebra is left adjoint 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.
If R is commutative and X is any set then A = RX = { f : X → R } with the pointwise operations is a commutative algebra.
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 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 X, the continuous linear operators form an associative algebra (using composition of operators as multiplication); this is a Banach algebra. Given any topological space 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 algebras, which are useful in geometry and physics. Incidence algebras of locally finite partially ordered sets are associative algebras considered in combinatorics. The partition algebra and its subalgebras, including the Brauer algebra and the Temperley-Lieb algebra. A differential graded algebra 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 so that the Lie bracket satisfies the Leibniz rule; i.e., . Given a Poisson algebra , consider the vector space of formal power series over . If has a structure of an associative algebra with multiplication such that, for ,
then is called a deformation quantization of . A quantized enveloping algebra. The dual of such an algebra turns out to be an associative algebra (see ) and is, philosophically speaking, the (quantized) coordinate ring of a quantum group. Gerstenhaber algebra
Constructions Subalgebras A subalgebra of an R-algebra A is a subset of A which is both a subring and a submodule of A. That is, it must be closed under addition, ring multiplication, scalar multiplication, and it must contain the identity element of A. Quotient algebras Let A be an R-algebra. Any ring-theoretic ideal I in A is automatically an R-module since . This gives the quotient ring the structure of an R-module and, in fact, an R-algebra. It follows that any ring homomorphic image of A is also an R-algebra. Direct products The direct product of a family of R-algebras is the ring-theoretic direct product. This becomes an R-algebra with the obvious scalar multiplication. Free products One can form a free product of R-algebras in a manner similar to the free product of groups. The free product is the coproduct in the category of R-algebras. Tensor products The tensor product of two R-algebras is also an R-algebra in a natural way. See tensor product of algebras for more details. Given a commutative ring R and any ring A the tensor product R ⊗Z A can be given the structure of an R-algebra by defining . The functor which sends A to is left adjoint to the functor which sends an R-algebra to its underlying ring (forgetting the module structure). See also: Change of rings. Free algebra A free algebra is an algebra generated by symbols. If one imposes commutativity; i.e., take the quotient by commutators, then one gets a polynomial algebra.
Dual of an associative algebra Let A be an associative algebra over a commutative ring R. Since A is in particular a module, we can take the dual module A* of A. A priori, the dual A* need not have a structure of an associative algebra. However, A may come with an extra structure (namely, that of a Hopf algebra) so that the dual is also an associative algebra.