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In mathematics, an associative algebra A is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field 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 vector space over K. In this article we will also use the term Kalgebra to mean an associative algebra over the field K. A standard first example of a Kalgebra is a ring of square matrices over a field K, with the usual matrix multiplication.
A commutative algebra is an associative algebra that has a commutative multiplication, 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 nonunital associative algebras. We will also assume that all rings are unital, and all ring homomorphisms are unital.
Many authors consider the more general concept of an associative algebra over a commutative ring R, instead of a field: An Ralgebra is an Rmodule with an associative Rbilinear binary operation, which also contains a multiplicative identity. For examples of this concept, if S is any ring with center C, then S is an associative Calgebra.
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Definition
Let R be a commutative ring (so R could be a field). An associative Ralgebra (or more simply, an Ralgebra) is a ring that is also an Rmodule 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 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 § From ring homomorphisms below).
Every ring is an associative algebra, where denotes the ring of the integers.
A commutative algebra 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 Ralgebra is a monoid object in RMod (the monoidal category of Rmodules). 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 Rbilinear map) corresponds to a unique Rlinear 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 Ralgebra by defining
for all r ∈ R and x ∈ A. If A is an Ralgebra, taking x = 1, 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 Ralgebra 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 R → A; i.e., commutative Ralgebras and whose morphisms are ring homomorphisms A → A' that are under R; i.e., R → A → A' is R → A' (i.e., the coslice category of the category of commutative rings under R.) The prime spectrum functor Spec then determines an antiequivalence 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 Ralgebras is an Rlinear ring homomorphism. Explicitly, is an associative algebra homomorphism if
The class of all Ralgebras together with algebra homomorphisms between them form a category, sometimes denoted RAlg.
The subcategory of commutative Ralgebras 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 Zalgebra. 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 Zalgebras are equivalent concepts, in the same way that abelian groups and Zmodules are equivalent.
 Any ring of characteristic n is a (Z/nZ)algebra in the same way.
 Given an Rmodule M, the endomorphism ring of M, denoted End_{R}(M) is an Ralgebra by defining (r·φ)(x) = r·φ(x).
 Any ring of matrices with coefficients in a commutative ring R forms an Ralgebra under matrix addition and multiplication. This coincides with the previous example when M is a finitelygenerated, free Rmodule.
 In particular, the square nbyn matrices with entries from the field K form an associative algebra over K.
 The complex numbers form a 2dimensional commutative algebra over the real numbers.
 The quaternions form a 4dimensional 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 R[x_{1}, ..., x_{n}] is a commutative Ralgebra. In fact, this is the free commutative Ralgebra on the set {x_{1}, ..., x_{n}}.
 The free Ralgebra 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 Rmodule is naturally an associative Ralgebra. The same is true for quotients such as the exterior and symmetric algebras. Categorically speaking, the functor that maps an Rmodule to its tensor algebra is left adjoint to the functor that sends an Ralgebra to its underlying Rmodule (forgetting the multiplicative structure).
 The following ring is used in the theory of λrings. Given a commutative ring A, let the set of formal power series with constant term 1. It is an abelian group with the group operation that is the multiplication of power series. It is then a ring with the multiplication, denoted by , such that determined by this condition and the ring axioms. The additive identity is 1 and the multiplicative identity is . Then has a canonical structure of a algebra given by the ring homomorphism On the other hand, if A is a λring, then there is a ring homomorphismgiving a structure of an Aalgebra.
 Given a module M over a commutative ring R, the direct sum of modules has a structure of an Ralgebra 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 quasifree 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 Ralgebra 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 A : X → X form an associative algebra (using composition of operators as multiplication); this is a Banach algebra.
 Given any topological space X, the continuous real or complexvalued 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 (Ω, F, (F_{t})_{t ≥ 0}, P) 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 TemperleyLieb 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 pforms 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 § Dual of an associative algebra) and is, philosophically speaking, the (quantized) coordinate ring of a quantum group.
 Gerstenhaber algebra
Constructions
 Subalgebras
 A subalgebra of an Ralgebra 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 Ralgebra. Any ringtheoretic ideal I in A is automatically an Rmodule since r · x = (r1_{A})x. This gives the quotient ring A / I the structure of an Rmodule and, in fact, an Ralgebra. It follows that any ring homomorphic image of A is also an Ralgebra.
 Direct products
 The direct product of a family of Ralgebras is the ringtheoretic direct product. This becomes an Ralgebra with the obvious scalar multiplication.
 Free products
 One can form a free product of Ralgebras in a manner similar to the free product of groups. The free product is the coproduct in the category of Ralgebras.
 Tensor products
 The tensor product of two Ralgebras is also an Ralgebra 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 Ralgebra by defining r · (s ⊗ a) = (rs ⊗ a). The functor which sends A to R ⊗_{Z} A is left adjoint to the functor which sends an Ralgebra 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.
For example, take A to be the ring of continuous functions on a compact group G. Then, not only A is an associative algebra, but it also comes with the comultiplication and counit .^{[1]} The "co" refers to the fact that they satisfy the dual of the usual multiplication and unit in the algebra axiom. Hence, the dual is an associative algebra. The comultiplication and counit are also important in order to form a tensor product of representations of associative algebras (see § Representations below).
Enveloping algebra
Given an associative algebra A over a commutative ring R, the enveloping algebra of A is the algebra or , depending on authors.^{[2]}
Note that a bimodule over A is exactly a left module over .
Separable algebra
Let A be an algebra over a commutative ring R. Then the algebra A is a right^{[3]} module over with the action . Then, by definition, A is said to separable if the multiplication map splits as an linear map,^{[4]} where is an module by . Equivalently,^{[5]} is separable if it is a projective module over ; thus, the projective dimension of A, sometimes called the bidimension of A, measures the failure of separability.
Finitedimensional algebra
Let A be a finitedimensional algebra over a field k. Then A is an Artinian ring.
Commutative case
As A is Artinian, if it is commutative, then it is a finite product of Artinian local rings whose residue fields are algebras over the base field k. Now, a reduced Artinian local ring is a field and thus the following are equivalent^{[6]}
 is separable.
 is reduced, where is some algebraic closure of k.
 for some n.
 is the number of algebra homomorphisms .
Let , the profinite group of finite Galois extensions of k. Then is an antiequivalence of the category of finitedimensional separable kalgebras to the category of finite sets with continuous actions.^{[7]}
Noncommutative case
Since a simple Artinian ring is a (full) matrix ring over a division ring, if A is a simple algebra, then A is a (full) matrix algebra over a division algebra D over k; i.e., . More generally, if A is a semisimple algebra, then it is a finite product of matrix algebras (over various division kalgebras), the fact known as the Artin–Wedderburn theorem.
The fact that A is Artinian simplifies the notion of a Jacobson radical; for an Artinian ring, the Jacobson radical of A is the intersection of all (twosided) maximal ideals (in contrast, in general, a Jacobson radical is the intersection of all left maximal ideals or the intersection of all right maximal ideals.)
The Wedderburn principal theorem states:^{[8]} for a finitedimensional algebra A with a nilpotent ideal I, if the projective dimension of as a module over the enveloping algebra is at most one, then the natural surjection splits; i.e., contains a subalgebra such that is an isomorphism. Taking I to be the Jacobson radical, the theorem says in particular that the Jacobson radical is complemented by a semisimple algebra. The theorem is an analog of Levi's theorem for Lie algebras.
Lattices and orders
Let R be a Noetherian integral domain with field of fractions K (for example, they can be ). A lattice L in a finitedimensional Kvector space V is a finitely generated Rsubmodule of V that spans V; in other words, .
Let be a finitedimensional Kalgebra. An order in is an Rsubalgebra that is a lattice. In general, there are a lot fewer orders than lattices; e.g., is a lattice in but not an order (since it is not an algebra).^{[9]}
A maximal order is an order that is maximal among all the orders.
Related concepts
Coalgebras
An associative algebra over K is given by a Kvector space A endowed with a bilinear map A × A → A having two inputs (multiplicator and multiplicand) and one output (product), as well as a morphism K → A identifying the scalar multiples of the multiplicative identity. If the bilinear map A × A → A is reinterpreted as a linear map (i. e., morphism in the category of Kvector spaces) A ⊗ A → A (by the universal property of the tensor product), then we can view an associative algebra over K as a Kvector space A endowed with two morphisms (one of the form A ⊗ A → A and one of the form K → A) satisfying certain conditions that boil down to the algebra axioms. These two morphisms can be dualized using categorial duality by reversing all arrows in the commutative diagrams that describe the algebra axioms; this defines the structure of a coalgebra.
There is also an abstract notion of Fcoalgebra, where F is a functor. This is vaguely related to the notion of coalgebra discussed above.
Representations
A representation of an algebra A is an algebra homomorphism ρ : A → End(V) from A to the endomorphism algebra of some vector space (or module) V. The property of ρ being an algebra homomorphism means that ρ preserves the multiplicative operation (that is, ρ(xy) = ρ(x)ρ(y) for all x and y in A), and that ρ sends the unit of A to the unit of End(V) (that is, to the identity endomorphism of V).
If A and B are two algebras, and ρ : A → End(V) and τ : B → End(W) are two representations, then there is a (canonical) representation A B → End(V W) of the tensor product algebra A B on the vector space V W. However, there is no natural way of defining a tensor product of two representations of a single associative algebra in such a way that the result is still a representation of that same algebra (not of its tensor product with itself), without somehow imposing additional conditions. Here, by tensor product of representations, the usual meaning is intended: the result should be a linear representation of the same algebra on the product vector space. Imposing such additional structure typically leads to the idea of a Hopf algebra or a Lie algebra, as demonstrated below.
Motivation for a Hopf algebra
Consider, for example, two representations and . One might try to form a tensor product representation according to how it acts on the product vector space, so that
However, such a map would not be linear, since one would have
for k ∈ K. One can rescue this attempt and restore linearity by imposing additional structure, by defining an algebra homomorphism Δ: A → A ⊗ A, and defining the tensor product representation as
Such a homomorphism Δ is called a comultiplication if it satisfies certain axioms. The resulting structure is called a bialgebra. To be consistent with the definitions of the associative algebra, the coalgebra must be coassociative, and, if the algebra is unital, then the coalgebra must be counital as well. A Hopf algebra is a bialgebra with an additional piece of structure (the socalled antipode), which allows not only to define the tensor product of two representations, but also the Hom module of two representations (again, similarly to how it is done in the representation theory of groups).
Motivation for a Lie algebra
One can try to be more clever in defining a tensor product. Consider, for example,
so that the action on the tensor product space is given by
 .
This map is clearly linear in x, and so it does not have the problem of the earlier definition. However, it fails to preserve multiplication:
 .
But, in general, this does not equal
 .
This shows that this definition of a tensor product is too naive; the obvious fix is to define it such that it is antisymmetric, so that the middle two terms cancel. This leads to the concept of a Lie algebra.
Nonunital algebras
Some authors use the term "associative algebra" to refer to structures which do not necessarily have a multiplicative identity, and hence consider homomorphisms which are not necessarily unital.
One example of a nonunital associative algebra is given by the set of all functions f: R → R whose limit as x nears infinity is zero.
Another example is the vector space of continuous periodic functions, together with the convolution product.
See also
 Abstract algebra
 Algebraic structure
 Algebra over a field
 Sheaf of algebras, a sort of an algebra over a ringed space
 Deligne's conjecture on Hochschild cohomology
Notes
 ^ Example 1 in Tjin, T. (October 10, 1992). "An introduction to quantized Lie groups and algebras". International Journal of Modern Physics A. 07 (25): 6175–6213. arXiv:hepth/9111043. Bibcode:1992IJMPA...7.6175T. doi:10.1142/S0217751X92002805. ISSN 0217751X. S2CID 119087306.
 ^ Vale 2009, Definition 3.1.
 ^ Editorial note: as it turns, is a full matrix ring in interesting cases and it is more conventional to let matrices act from the right.
 ^ Cohn 2003, § 4.7.
 ^ To see the equivalence, note a section of can be used to construct a section of a surjection.
 ^ Waterhouse 1979, § 6.2.
 ^ Waterhouse 1979, § 6.3.
 ^ Cohn 2003, Theorem 4.7.5.
 ^ Artin 1999, Ch. IV, § 1.
References
 Artin, Michael (1999). "Noncommutative Rings" (PDF). Archived (PDF) from the original on October 9, 2022.
 Bourbaki, N. (1989). Algebra I. Springer. ISBN 3540642439.
 Cohn, P.M. (2003). Further Algebra and Applications (2nd ed.). Springer. ISBN 1852336676. Zbl 1006.00001.
 Nathan Jacobson, Structure of Rings
 James Byrnie Shaw (1907) A Synopsis of Linear Associative Algebra, link from Cornell University Historical Math Monographs.
 Ross Street (1998) Quantum Groups: an entrée to modern algebra, an overview of indexfree notation.
 Vale, R. (2009). "notes on quasifree algebras" (PDF).
 Waterhouse, William (1979), Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Berlin, New York: SpringerVerlag, doi:10.1007/9781461262176, ISBN 9780387904214, MR 0547117