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In [[mathematics]], an '''associative algebra''' ''A'' over a [[commutative ring]] (often a [[Field (mathematics)|field]]) ''K'' is a [[ring (mathematics)|ring]] ''A'' together with a [[ring homomorphism]] from ''K'' into the [[center (ring theory)|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 (mathematics)|ring]]; the addition and scalar multiplication operations together give ''A'' the structure of a [[module (mathematics)|module]] or [[vector space]] over ''K''. In this article we will also use the term [[algebra over a field|''K''-algebra]] to mean an associative algebra over  ''K''. A standard first example of a ''K''-algebra is a ring of [[Square matrix|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 [[unital algebra|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.

{{Algebraic structures |Algebra}}

== 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 (mathematics)|ring]] ''A'' 
that is also an [[module (mathematics)|''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
: <math>r\cdot(xy) = (r\cdot x)y = x(r\cdot y)</math>
for all ''r'' in ''R'' and ''x'', ''y'' in the algebra. (This definition implies that the algebra, being a ring, is [[unital algebra|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 (ring theory)|center]] of ''A''. If ''f'' is such a homomorphism, the scalar multiplication is {{nowrap|(''r'', ''x'') ↦ ''f''(''r'')''x''}} (here the multiplication is the ring multiplication); if the scalar multiplication is given, the ring homomorphism is given by {{nowrap|''r'' ↦ ''r'' ⋅ 1<sub>''A''</sub>}}. (See also ''{{slink|#From ring homomorphisms}}'' below).

Every ring is an associative '''Z'''-algebra, where '''Z''' denotes the ring of the [[integer]]s.

A '''{{vanchor|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 ''R''-algebra is a [[monoid (category theory)|monoid object]] in [[category of modules|'''''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
: <math>m : A \otimes_R A \to A</math>.
The associativity then refers to the identity:
: <math>m \circ ({\operatorname{id}} \otimes m) = m \circ (m \otimes \operatorname{id}).</math>

=== From ring homomorphisms ===
An associative algebra amounts to a [[ring homomorphism]] whose image lies in the [[center of a ring|center]]. Indeed, starting with a ring ''A'' and a ring homomorphism {{nowrap|''η'' : ''R'' → ''A''}} whose image lies in the [[center (ring theory)|center]] of ''A'', we can make ''A'' an ''R''-algebra by defining
: <math>r\cdot x = \eta(r)x</math>
for all {{nowrap|''r'' ∈ ''R''}} and {{nowrap|''x'' ∈ ''A''}}. If ''A'' is an ''R''-algebra, taking {{nowrap|1=''x'' = 1}}, the same formula in turn defines a ring homomorphism {{nowrap|''η'' : ''R'' → ''A''}} 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 {{nowrap|''η'' : ''R'' → ''A''}}.

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 {{nowrap|''R'' → ''A''}} for a fixed ''R'', i.e., commutative ''R''-algebras, and whose morphisms are ring homomorphisms {{nowrap|''A'' → ''A''′}} that are under ''R''; i.e., {{nowrap|''R'' → ''A'' → ''A''′}} is {{nowrap|''R'' → ''A''′}} (i.e., the [[coslice category]] of the category of commutative rings under ''R''.)  The [[prime spectrum]] functor Spec then determines an [[dual (category theory)|anti-equivalence]] of this category to the category of [[affine scheme]]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 ==
{{main|algebra homomorphism}}
A [[homomorphism]] between two ''R''-algebras is an [[module homomorphism|''R''-linear]] [[ring homomorphism]]. Explicitly, {{nowrap|''φ'' : ''A''<sub>1</sub> → ''A''<sub>2</sub>}} is an '''associative algebra homomorphism''' if
: <math>\begin{align}
  \varphi(r \cdot x) &= r \cdot \varphi(x) \\
      \varphi(x + y) &= \varphi(x) + \varphi(y) \\
         \varphi(xy) &= \varphi(x)\varphi(y) \\
          \varphi(1) &= 1
\end{align}</math>

The class of all ''R''-algebras together with algebra homomorphisms between them form a [[category (mathematics)|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 (ring theory)|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]]s and '''Z'''-modules are equivalent.
* Any ring of [[characteristic (algebra)|characteristic]] ''n'' is a ('''Z'''/''n'''''Z''')-algebra in the same way.
* Given an ''R''-module ''M'', the [[endomorphism ring]] of ''M'', denoted End<sub>''R''</sub>(''M'') is an ''R''-algebra by defining {{nowrap|1=(''r''·''φ'')(''x'') = ''r''·''φ''(''x'')}}.
* Any ring of [[matrix (mathematics)|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 module|free]] ''R''-module.
** In particular, the square ''n''-by-''n'' [[square matrix|matrices]] with entries from the field ''K'' form an associative algebra over ''K''.
* The [[complex number]]s form a 2-dimensional commutative algebra over the [[real number]]s.
* The [[quaternion]]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]] {{nowrap|''R''[''x''<sub>1</sub>, ..., ''x<sub>n</sub>'']}} is a commutative ''R''-algebra. In fact, this is the free commutative ''R''-algebra on the set {{nowrap|{{mset|''x''<sub>1</sub>, ..., ''x<sub>n</sub>''}}}}.
* The [[free algebra|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 algebra|exterior]] and [[symmetric algebra]]s. 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 {{nowrap|1=''R'' ⊕ ''M''}} has a structure of an ''R''-algebra by thinking ''M'' consists of infinitesimal elements; i.e., the multiplication is given as {{nowrap|1=(''a'' + ''x'')(''b'' + ''y'') = ''ab'' + ''ay'' + ''bx''}}. 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 ring|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 function (topology)|continuous]] [[linear operator]]s {{nowrap|''A'' : ''X'' → ''X''}} form an associative algebra (using composition of operators as multiplication); this is a [[Banach algebra]].
* Given any [[topology|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 [[semimartingale]]s defined on the [[filtration (mathematics)#Measure theory|filtered probability space]] {{nowrap|(Ω, ''F'', (''F''<sub>''t''</sub>)<sub>''t''≥0</sub>, P)}} forms a ring under [[stochastic calculus|stochastic integration]].{{citation needed|date=October 2023}}
* The [[Weyl algebra]]
* An [[Azumaya algebra]]

=== Geometry and combinatorics ===
* The [[Clifford algebra]]s, which are useful in [[geometry]] and [[physics]].
* [[Incidence algebra]]s of [[locally finite poset|locally finite]] [[partially ordered set]]s 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]] <math display="inline">\Omega(M) = \bigoplus_{p=0}^n \Omega^p(M)</math>, where <math display="inline">\Omega^p(M)</math> 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 {{mset|,}} satisfies the Leibniz rule; i.e., {{nowrap|1={{mset|''fg'', ''h''}} = {{itco|''f''}}{{mset|''g'', ''h''}} + ''g''{{mset|''f'', ''h''}}}}.
* Given a Poisson algebra <math>\mathfrak a</math>, consider the vector space <math>\mathfrak{a}[\![u]\!]</math> of [[formal power series]] over <math>\mathfrak{a}</math>.  If  <math>\mathfrak{a}[\![u]\!]</math> has a structure of an associative algebra with multiplication <math>*</math> such that, for <math>f, g \in \mathfrak{a}</math>,
*: <math>f * g = f g - \frac{1}{2} \{ f, g \} u + \cdots,</math>
: then <math>\mathfrak{a}[\![u]\!]</math> is called a [[deformation quantization]] of <math>\mathfrak a</math>.
* A [[quantized enveloping algebra]]. The dual of such an algebra turns out to be an associative algebra (see {{slink||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 ''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 (ring theory)|ideal]] ''I'' in ''A'' is automatically an ''R''-module since {{nowrap|1=''r'' · ''x'' = (''r''1<sub>''A''</sub>)''x''}}. This gives the [[quotient ring]] {{nowrap|''A'' / ''I''}} 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 [[product of rings|direct product]]. This becomes an ''R''-algebra with the obvious scalar multiplication.
; Free products: One can form a [[free product of associative algebras|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 of rings|tensor product]] ''R''&nbsp;⊗<sub>'''Z'''</sub>&nbsp;''A'' can be given the structure of an ''R''-algebra by defining {{nowrap|1=''r'' · (''s'' ⊗ ''a'') = (''rs'' ⊗ ''a'')}}. The functor which sends ''A'' to {{nowrap|''R'' ⊗<sub>'''Z'''</sub> ''A''}} 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''<sup>*</sup> of ''A''. A priori, the dual ''A''<sup>*</sup> 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 co-multiplication {{nowrap|1=Δ({{itco|''f''}})(''g'', ''h'') = {{itco|''f''}}(''gh'')}} and co-unit {{nowrap|1=''ε''({{itco|''f''}}) = {{itco|''f''}}(1)}}.{{sfn|Tjin|1992|loc=Example 1|ps=none}} The "co-" refers to the fact that they satisfy the dual of the usual multiplication and unit in the algebra axiom. Hence, the dual ''A''<sup>*</sup> is an associative algebra. The co-multiplication and co-unit are also important in order to form a tensor product of representations of associative algebras (see ''{{slink|#Representations}}'' below).

== Enveloping algebra ==
{{expand section|date=March 2023}}
{{see also|Non-associative algebra#Associated algebras}}

Given an associative algebra ''A'' over a commutative ring ''R'', the '''enveloping algebra''' ''A''<sup>e</sup> of ''A'' is the algebra {{nowrap|''A'' ⊗<sub>''R''</sub> ''A''<sup>op</sup>}} or {{nowrap|''A''<sup>op</sup> ⊗<sub>''R''</sub> ''A''}}, depending on authors.{{sfn|Vale|2009|loc=Definition 3.1|ps=none}}

Note that a [[bimodule]] over ''A'' is exactly a left module over ''A''<sup>e</sup>.

== Separable algebra ==
{{main|Separable algebra}}

Let ''A'' be an algebra over a commutative ring ''R''. Then the algebra ''A'' is a right{{efn|Editorial note: as it turns out, ''A''<sup>e</sup> is a full matrix ring in interesting cases and it is more conventional to let matrices act from the right.}} module over {{nowrap|1=''A''<sup>e</sup> := ''A''<sup>op</sup> ⊗<sub>''R''</sub> ''A''}} with the action {{nowrap|1=''x'' ⋅ (''a'' ⊗ ''b'') = ''axb''}}. Then, by definition, ''A'' is said to [[separable algebra|separable]] if the multiplication map {{nowrap|''A'' ⊗<sub>''R''</sub> ''A'' → ''A'' : ''x'' ⊗ ''y'' ↦ ''xy''}} splits as an ''A''<sup>e</sup>-linear map,{{sfn|Cohn|2003|loc=§ 4.7|ps=none}} where {{nowrap|''A'' ⊗ ''A''}} is an ''A''<sup>e</sup>-module by {{nowrap|1=(''x'' ⊗ ''y'') ⋅ (''a'' ⊗ ''b'') = ''ax'' ⊗ ''yb''}}. Equivalently,{{efn|To see the equivalence, note a section of {{nowrap|''A'' ⊗<sub>''R''</sub> ''A'' → ''A''}} can be used to construct a section of a surjection.}}
''A'' is separable if it is a [[projective module]] over {{nowrap|''A''<sup>e</sup>}}; thus, the {{nowrap|''A''<sup>e</sup>}}-projective dimension of ''A'', sometimes called the '''bidimension''' of ''A'', measures the failure of separability.

== Finite-dimensional algebra ==
{{See also|Central simple algebra}}

Let ''A'' be a finite-dimensional 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 field]]s are algebras over the base field ''k''. Now, a [[reduced ring|reduced]] Artinian local ring is a field and thus the following are equivalent{{sfn|Waterhouse|1979|loc=§ 6.2|ps=none}}
# <math>A</math> is separable.
# <math>A \otimes \overline{k}</math> is reduced, where <math>\overline{k}</math> is some [[algebraic closure]] of ''k''.
# <math>A \otimes \overline{k} = \overline{k}^n</math> for some ''n''.
# <math>\dim_k A</math> is the number of <math>k</math>-algebra homomorphisms <math>A \to \overline{k}</math>.

Let <math>\Gamma = \operatorname{Gal}(k_s/k) = \varprojlim \operatorname{Gal}(k'/k)</math>, the [[profinite group]] of finite Galois extensions of ''k''. Then <math>A \mapsto X_A = \{ k\text{-algebra homomorphisms } A \to k_s \}</math> is an anti-equivalence of the category of finite-dimensional separable ''k''-algebras to the category of finite sets with continuous <math>\Gamma</math>-actions.{{sfn|Waterhouse|1979|loc=§ 6.3|ps=none}}

=== 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., {{nowrap|1=''A'' = M<sub>''n''</sub>(''D'')}}. More generally, if ''A'' is a semisimple algebra, then it is a finite product of matrix algebras (over various division ''k''-algebras), 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 (two-sided) 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.)

{{anchor|Wedderburn principal theorem}}
The '''Wedderburn principal theorem''' states:{{sfn|Cohn|2003|loc=Theorem 4.7.5|ps=none}} for a finite-dimensional algebra ''A'' with a [[nilpotent ideal]] ''I'', if the projective dimension of {{nowrap|''A'' / ''I''}} as a module over the [[enveloping algebra of an associative algebra|enveloping algebra]] {{nowrap|(''A'' / ''I'')<sup>e</sup>}} is at most one, then the natural surjection {{nowrap|''p'' : ''A'' → ''A'' / ''I''}} splits; i.e., ''A'' contains a subalgebra ''B'' such that {{nowrap|''p''{{!}}<sub>''B''</sub> : ''B'' {{overset|lh=0.5|~|→}} ''A'' / ''I''}} 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 algebra]]s.<!--
A finite-dimensional algebra ''A'' is called a [[split algebra]] if each endomorphism of a simple ''A''-module is given by a scalar multiplication. Equivalently,

For example, a finite-dimensional algebra is a split when the base field is algebraically closed.-->

== Lattices and orders ==
{{main|Order (ring theory)}}
Let ''R'' be a [[Noetherian ring|Noetherian]] [[integral domain]] with field of fractions ''K'' (for example, they can be '''Z''', '''Q'''). A ''[[lattice (module)|lattice]]'' ''L'' in a finite-dimensional ''K''-vector space ''V'' is a finitely generated ''R''-submodule of ''V'' that spans ''V''; in other words, {{nowrap|1=''L'' ⊗<sub>''R''</sub> ''K'' = ''V''}}.

Let ''A''<sub>''K''</sub> be a finite-dimensional ''K''-algebra. An ''[[order (ring theory)|order]]'' in ''A''<sub>''K''</sub> is an ''R''-subalgebra that is a lattice. In general, there are a lot fewer orders than lattices; e.g., {{sfrac|1|2}}'''Z''' is a lattice in '''Q''' but not an order (since it is not an algebra).{{sfn|Artin|1999|loc=Ch. IV, § 1|ps=none}}

A ''maximal order'' is an order that is maximal among all the orders.

== Related concepts ==
=== Coalgebras ===
{{Main|Coalgebra}}
An associative algebra over ''K'' is given by a ''K''-vector space ''A'' endowed with a bilinear map {{nowrap|''A'' × ''A'' → ''A''}} having two inputs (multiplicator and multiplicand) and one output (product), as well as a morphism {{nowrap|''K'' → ''A''}} identifying the scalar multiples of the multiplicative identity. If the bilinear map {{nowrap|''A'' × ''A'' → ''A''}} is reinterpreted as a linear map (i.e., [[morphism]] in the category of ''K''-vector spaces) {{nowrap|''A'' ⊗ ''A'' → ''A''}} (by the [[Tensor product#Characterization by a universal property|universal property of the tensor product]]), then we can view an associative algebra over ''K'' as a ''K''-vector space ''A'' endowed with two morphisms (one of the form {{nowrap|''A'' ⊗ ''A'' → ''A''}} and one of the form {{nowrap|''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 diagram]]s that describe the algebra [[axiom]]s; this defines the structure of a [[coalgebra]].

There is also an abstract notion of [[F-coalgebra|''F''-coalgebra]], where ''F'' is a [[functor]]. This is vaguely related to the notion of coalgebra discussed above.

== Representations ==
{{main|Algebra representation}}
A [[representation theory|representation]] of an algebra ''A'' is an algebra homomorphism {{nowrap|''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, {{nowrap|1=''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 {{nowrap|''ρ'' : ''A'' → End(''V'')}} and {{nowrap|''τ'' : ''B'' → End(''W'')}} are two representations, then there is a (canonical) representation {{nowrap|''A'' ⊗ ''B'' → End(''V'' ⊗ ''W'')}} of the tensor product algebra {{nowrap|''A'' ⊗ ''B''}} on the vector space {{nowrap|''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 {{nowrap|''σ'' : ''A'' → End(''V'')}} and {{nowrap|''τ'' : ''A'' → End(''W'')}}.  One might try to form a tensor product representation {{nowrap|''ρ'' : ''x'' ↦ ''σ''(''x'') ⊗ ''τ''(''x'')}} according to how it acts on the product vector space, so that
: <math>\rho(x)(v \otimes w) = (\sigma(x)(v)) \otimes (\tau(x)(w)).</math>
However, such a map would not be linear, since one would have
: <math>\rho(kx) = \sigma(kx) \otimes \tau(kx) = k\sigma(x) \otimes k\tau(x) = k^2 (\sigma(x) \otimes \tau(x)) = k^2 \rho(x)</math>
for {{nowrap|''k'' ∈ ''K''}}. One can rescue this attempt and restore linearity by imposing additional structure, by defining an algebra homomorphism {{nowrap|Δ : ''A'' → ''A'' ⊗ ''A''}}, and defining the tensor product representation as
: <math>\rho = (\sigma\otimes \tau) \circ \Delta.</math>
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 co-associative, and, if the algebra is unital, then the co-algebra must be co-unital as well. A [[Hopf algebra]] is a bialgebra with an additional piece of structure (the so-called 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 ===
{{See also|Lie algebra representation}}
One can try to be more clever in defining a tensor product. Consider, for example,
: <math>x \mapsto \rho (x) = \sigma(x) \otimes \mbox{Id}_W + \mbox{Id}_V \otimes \tau(x)</math>
so that the action on the tensor product space is given by
: <math>\rho(x) (v \otimes w) = (\sigma(x) v)\otimes w + v \otimes (\tau(x) w) </math>.

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:
: <math>\rho(xy) = \sigma(x) \sigma(y) \otimes \mbox{Id}_W + \mbox{Id}_V \otimes \tau(x) \tau(y)</math>.
But, in general, this does not equal
: <math>\rho(x)\rho(y) = \sigma(x) \sigma(y) \otimes \mbox{Id}_W + \sigma(x) \otimes \tau(y) + \sigma(y) \otimes \tau(x) + \mbox{Id}_V \otimes \tau(x) \tau(y)</math>.
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]].

== Non-unital 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 non-unital associative algebra is given by the set of all functions {{nowrap|''f'' : '''R''' → '''R'''}} whose [[limit of a function|limit]] as ''x'' nears infinity is zero.

Another example is the vector space of continuous periodic functions, together with the [[convolution|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 ==
{{notelist}}

== Citations ==
{{reflist}}

== References ==
{{refbegin}}
* {{cite web
  |last1=Artin |first1=Michael
  |year=1999
  |title=Noncommutative Rings
  |url=http://math.mit.edu/~etingof/artinnotes.pdf
  |archive-url=https://ghostarchive.org/archive/20221009/http://math.mit.edu/~etingof/artinnotes.pdf
  |archive-date=2022-10-09
  |url-status=live
  }}
* {{cite book
  |author=Bourbaki, N.
  |title=Algebra I
  |publisher=Springer
  |year=1989
  |isbn=3-540-64243-9
  |url=https://books.google.com/books?id=STS9aZ6F204C&q=%22associative+algebra%22
  }}
* {{cite book
  | title=Further Algebra and Applications
  | last1=Cohn | first1=P.M. | author-link=Paul Cohn
  | year=2003
  | edition=2nd
  | publisher=Springer
  | isbn=1852336676
  | zbl=1006.00001
  }}
* {{citation
  |last1=Jacobson |first1=Nathan
  |year=1956
  |title=Structure of Rings
  |series=Colloquium Publications |volume=37
  |publisher=American Mathematical Society
  |isbn=978-0-8218-1037-8
 |url=https://books.google.com/books?id=KwviDgAAQBAJ
  }}
* {{cite book|last=Shaw|first=James Byrnie|year=1907|url=https://ebooks.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=05160001|title=A Synopsis of Linear Associative Algebra|publisher=[[Cornell University]] Historical Math Monographs}}
* {{cite book|last=Street|first=Ross|year=1998|url=https://web.archive.org/web/20050825034431/http://www-texdev.ics.mq.edu.au/Quantum/Quantum.ps|title=Quantum Groups: an entrée to modern algebra}} An overview of index-free notation.
* {{cite journal
  |last1=Tjin |first1=T.
  |title=An introduction to quantized Lie groups and algebras
  |journal=International Journal of Modern Physics A
  |date=10 October 1992
  |volume=07 |issue=25
  |pages=6175–6213
  |doi=10.1142/S0217751X92002805
  |arxiv=hep-th/9111043
  |bibcode=1992IJMPA...7.6175T
  |s2cid=119087306
  |issn=0217-751X
  }}
* {{cite web
  |last1=Vale |first1=R.
  |date=2009
  |title=notes on quasi-free algebras
  |url=https://pi.math.cornell.edu/~rvale/ada.pdf
  }}
* {{citation
  |last1=Waterhouse |first1=William
  |author1-link=William C. Waterhouse
  |year=1979
  |title=Introduction to affine group schemes
  |series=Graduate Texts in Mathematics |volume=66
  |publisher=[[Springer-Verlag]] |location=Berlin, New York
  |isbn=978-0-387-90421-4
  |doi=10.1007/978-1-4612-6217-6
  |mr=0547117
  }}
{{refend}}

{{Authority control}}

{{DEFAULTSORT:Associative Algebra}}
[[Category:Algebras]]
[[Category:Algebraic geometry]]