View source for Algebra over a field

{{Short description|Vector space equipped with a bilinear product}}
{{Algebraic structures |Algebra}}
In [[mathematics]], an '''algebra over a field''' (often simply called an '''algebra''') is a [[vector space]] equipped with a [[bilinear map|bilinear]] [[product (mathematics)|product]]. Thus, an algebra is an [[algebraic structure]] consisting of a [[set (mathematics)|set]] together with operations of multiplication and addition and [[scalar multiplication]] by elements of a [[field (mathematics)|field]] and satisfying the axioms implied by "vector space" and "bilinear".<ref>See also {{harvnb|Hazewinkel|Gubareni|Kirichenko|2004|p=[{{Google books|AibpdVNkFDYC|plainurl=y|page=3|text=an algebra over a field k}} 3] Proposition 1.1.1}}</ref>
The multiplication operation in an algebra may or may not be [[associative]], leading to the notions of [[associative algebra]]s and [[non-associative algebra]]s. Given an integer ''n'', the [[ring (mathematics)|ring]] of [[real matrix|real]] [[square matrix|square matrices]] of order ''n'' is an example of an associative algebra over the field of [[real number]]s under [[matrix addition]] and [[matrix multiplication]] since matrix multiplication is associative. Three-dimensional [[Euclidean space]] with multiplication given by the [[vector cross product]] is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the [[Jacobi identity]] instead.
An algebra is '''unital''' or '''unitary''' if it has an [[identity element]] with respect to the multiplication. The ring of real square matrices of order ''n'' forms a unital algebra since the [[identity matrix]] of order ''n'' is the identity element with respect to matrix multiplication. It is an example of a unital associative algebra, a [[unital ring|(unital) ring]] that is also a vector space.
Many authors use the term ''algebra'' to mean ''associative algebra'', or ''unital associative algebra'', or in  some subjects such as [[algebraic geometry]], ''unital associative commutative algebra''.
Replacing the field of scalars by a [[commutative ring]] leads to the more general notion of an [[#Generalization: algebra over a ring|algebra over a ring]]. Algebras are not to be confused with vector spaces equipped with a [[bilinear form]], like [[inner product space]]s, as, for such a space, the result of a product is not in the space, but rather in the field of coefficients.
== Definition and motivation ==
===Motivating examples===
{| class="wikitable"
|+
!Algebra
!vector space
!bilinear operator
![[associativity]]
![[commutativity]]
|-
|[[complex number]]s
|<math>\mathbb{R}^2</math>
|[[Complex number#Multiplication|product of complex numbers]]<br><math>\left(a+ib \right) \cdot \left(c+id \right)</math>
|Yes
|Yes
|-
|[[cross product]] of 3D vectors
|<math>\mathbb{R}^3</math>
|[[cross product]] <br><math>\vec{a} \times \vec{b}</math>
|No
|No ([[Anticommutative property|anticommutative]])
|-
|[[quaternion]]s
|<math>\mathbb{R}^4</math>
|[[Quaternion#Hamilton product|Hamilton product]] <br><math>(a+\vec{v})(b+\vec{w})</math>
|Yes
|No
|-
|[[polynomial]]s
|<math>\mathbb{R}[X]</math>
|[[polynomial multiplication]]
|Yes
|Yes
|-
| [[square matrix|square matrices]]
| <math>\mathbb R^{n\times n}</math>
| [[matrix multiplication]]
| Yes
| No
|}
=== Definition ===
Let {{mvar|K}} be a [[field (mathematics)|field]], and let {{mvar|A}} be a [[vector space]] over {{mvar|K}} equipped with an additional [[binary operation]] from {{math|''A'' × ''A''}} to {{mvar|A}}, denoted here by {{math|·}} (that is, if {{mvar|x}} and {{mvar|y}} are any two elements of {{mvar|A}}, then {{math|''x'' · ''y''}} is an element of {{mvar|A}} that is called the ''product'' of {{mvar|x}} and {{mvar|y}}).  Then {{mvar|A}} is an ''algebra'' over {{mvar|K}} if the following identities hold for all elements {{math|''x'', ''y'', ''z''}} in {{mvar|A}} , and all elements (often called [[scalar (mathematics)|scalar]]s) {{mvar|a}} and {{mvar|b}} in {{mvar|K}}:
* Right [[distributivity]]: {{math|1=(''x'' + ''y'') · ''z'' = ''x'' · ''z'' + ''y'' · ''z''}}
* Left distributivity: {{math|1=''z'' · (''x'' + ''y'') = ''z'' · ''x'' + ''z'' · ''y''}}
* Compatibility with scalars: {{math|1=(''ax'') · (''by'') = (''ab'') (''x'' · ''y'')}}.
These three axioms are another way of saying that the binary operation is [[bilinear operator|bilinear]]. An algebra over {{mvar|K}} is sometimes also called a ''{{mvar|K}}-algebra'', and {{mvar|K}} is called the ''base field'' of {{mvar|A}}. The binary operation is often referred to as ''multiplication'' in {{mvar|A}}. The convention adopted in this article is that multiplication of elements of an algebra is not necessarily [[associativity|associative]], although some authors use the term ''algebra'' to refer to an [[associative algebra]].
When a binary operation on a vector space is [[commutative]], left distributivity and right distributivity are equivalent, and, in this case, only one distributivity requires a proof. In general, for non-commutative operations left distributivity and right distributivity are not equivalent, and require separate proofs.
== Basic concepts ==
=== Algebra homomorphisms ===
Given {{math|''K''}}-algebras {{math|''A''}} and {{math|''B''}}, a [[homomorphism]] of {{math|''K''}}-algebras or {{math|''K''}}-'''algebra homomorphism''' is a {{math|''K''}}-[[linear map]] {{math|''f'': ''A'' → ''B''}} such that {{math|1=''f''(''xy'') = ''f''(''x'') ''f''(''y'')}} for all {{math|''x'', ''y''}} in {{math|''A''}}. If {{math|''A''}} and {{math|''B''}} are unital, then a homomorphism satisfying {{math|1=''f''(1{{sub|''A''}}) = 1{{sub|''B''}}}} is said to be a unital homomorphism. The space of all {{math|''K''}}-algebra homomorphisms between {{math|''A''}} and {{math|''B''}} is frequently written as
:<math>\mathbf{Hom}_{K\text{-alg}} (A,B).</math>
A {{math|''K''}}-algebra [[isomorphism]] is a [[bijective]] {{math|''K''}}-algebra homomorphism.
=== Subalgebras and ideals ===
{{main|Substructure (mathematics)}}
A ''subalgebra'' of an algebra over a field ''K'' is a [[linear subspace]] that has the property that the product of any two of its elements is again in the subspace. In other words, a subalgebra of an algebra is a non-empty subset of elements that is closed under addition, multiplication, and scalar multiplication. In symbols, we say that a subset ''L'' of a ''K''-algebra ''A'' is a subalgebra if for every ''x'', ''y'' in ''L'' and ''c'' in ''K'', we have that ''x'' · ''y'', ''x'' + ''y'', and ''cx'' are all in ''L''.
In the above example of the complex numbers viewed as a two-dimensional algebra over the real numbers, the one-dimensional real line is a subalgebra.
A ''left ideal'' of a ''K''-algebra is a linear subspace that has the property that any element of the subspace multiplied on the left by any element of the algebra produces an element of the subspace. In symbols, we say that a subset ''L'' of a ''K''-algebra ''A'' is a left ideal if for every ''x'' and ''y'' in ''L'', ''z'' in ''A'' and ''c'' in ''K'', we have the following three statements.
# ''x'' + ''y'' is in ''L'' (''L'' is closed under addition),
# ''cx'' is in ''L'' (''L'' is closed under scalar multiplication),
# ''z'' · ''x'' is in ''L'' (''L'' is closed under left multiplication by arbitrary elements).
If (3) were replaced with ''x'' · ''z'' is in ''L'', then this would define a ''right ideal''. A ''two-sided ideal'' is a subset that is both a left and a right ideal. The term ''ideal'' on its own is usually taken to mean a two-sided ideal. Of course when the algebra is commutative, then all of these notions of ideal are equivalent. Conditions (1) and (2) together are equivalent to ''L'' being a linear subspace of ''A''. It follows from condition (3) that every left or right ideal is a subalgebra.
This definition is different from the definition of an [[ideal (ring theory)|ideal of a ring]], in that here we require the condition (2). Of course if the algebra is unital, then condition (3) implies condition (2).
=== Extension of scalars ===
{{main|Extension of scalars}}
If we have a [[field extension]] ''F''/''K'', which is to say a bigger field ''F'' that contains ''K'', then there is a natural way to construct an algebra over ''F'' from any algebra over ''K''. It is the same construction one uses to make a vector space over a bigger field, namely the tensor product <math> V_F:=V \otimes_K F </math>. So if ''A'' is an algebra over ''K'', then <math>A_F</math> is an algebra over ''F''.
== Kinds of algebras and examples ==
Algebras over fields come in many different types. These types are specified by insisting on some further axioms, such as [[commutativity]] or [[associativity]] of the multiplication operation, which are not required in the broad definition of an algebra. The theories corresponding to the different types of algebras are often very different.
=== Unital algebra ===
An algebra is ''unital'' or ''unitary'' if it has a [[Unit (algebra)|unit]] or identity element ''I'' with ''Ix'' = ''x'' = ''xI'' for all ''x'' in the algebra.
=== Zero algebra ===
<!--The term '''zero algebra''' may have other uses outside ring theory-->
An algebra is called a '''zero algebra''' if {{nowrap|1=''uv'' = 0}} for all ''u'', ''v'' in the algebra,<ref>{{cite book |first=João B. |last=Prolla |title=Approximation of Vector Valued Functions |chapter=Lemma 4.10 |chapter-url=https://books.google.com/books?id=utTS4nTd-IsC |date=2011 |publisher=Elsevier |isbn=978-0-08-087136-3 |orig-year=1977 |page=65}}</ref> not to be confused with the algebra with one element.  It is inherently non-unital (except in the case of only one element), associative and commutative.
One may define a '''unital zero algebra''' by taking the [[direct sum of modules]] of a field (or more generally a ring) ''K'' and a ''K''-vector space (or module) ''V'', and defining the product of every pair of elements of ''V'' to be zero. That is, if {{nowrap|''λ'', ''μ'' ∈ ''K''}} and {{nowrap|''u'', ''v'' ∈ ''V''}}, then {{nowrap|1=(''λ'' + ''u'') (''μ'' + ''v'') = ''λμ'' + (''λv'' + ''μu'')}}. If {{nowrap|''e''<sub>1</sub>, ... ''e''<sub>''d''</sub>}} is a basis of ''V'', the unital zero algebra is the quotient of the polynomial ring {{nowrap|''K''[''E''<sub>1</sub>, ..., ''E''<sub>''n''</sub>]}} by the [[ideal (ring theory)|ideal]] generated by the ''E''<sub>''i''</sub>''E''<sub>''j''</sub> for every pair {{nowrap|(''i'', ''j'')}}.
An example of unital zero algebra is the algebra of [[dual number]]s, the unital zero '''R'''-algebra built from a one dimensional real vector space.
These unital zero algebras may be more generally useful, as they allow to translate any general property of the algebras to properties of vector spaces or [[module (mathematics)|modules]]. For example, the theory of [[Gröbner basis|Gröbner bases]] was introduced by [[Bruno Buchberger]] for [[ideal (ring theory)|ideals]] in a polynomial ring {{nowrap|1=''R'' = ''K''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>]}} over a field. The construction of the unital zero algebra over a free ''R''-module allows extending this theory as a Gröbner basis theory for submodules of a free module. This extension allows, for computing a Gröbner basis of a submodule, to use, without any modification, any algorithm and any software for computing Gröbner bases of ideals.
=== Associative algebra ===
{{main|Associative algebra}}
Examples of associative algebras include
* the algebra of all ''n''-by-''n'' [[matrix (mathematics)|matrices]] over a field (or commutative ring) ''K''. Here the multiplication is ordinary [[matrix multiplication]].
* [[Group ring|group algebra]]s, where a [[group (mathematics)|group]] serves as a basis of the vector space and algebra multiplication extends group multiplication.
* the commutative algebra ''K''[''x''] of all [[polynomial]]s over ''K'' (see [[polynomial ring]]).
* algebras of [[function (mathematics)|function]]s, such as the '''R'''-algebra of all real-valued [[continuous function|continuous]] functions defined on the [[interval (mathematics)|interval]] [0,1], or the '''C'''-algebra of all [[holomorphic function]]s defined on some fixed open set in the [[complex plane]]. These are also commutative.
* [[Incidence algebra]]s are built on certain [[partially ordered set]]s.
* algebras of [[linear operator]]s, for example on a [[Hilbert space]]. Here the algebra multiplication is given by the [[functional composition|composition]] of operators. These algebras also carry a [[topological space|topology]]; many of them are defined on an underlying [[Banach space]], which turns them into [[Banach algebra]]s. If an involution is given as well, we obtain [[B*-algebra]]s and [[C*-algebra]]s. These are studied in [[functional analysis]].
=== Non-associative algebra ===
{{main|Non-associative algebra}}
A ''non-associative algebra''<ref name=Schafer>{{cite book |first=Richard D. |last=Schafer|author-link=Richard D. Schafer|title=An Introduction to Nonassociative Algebras |year=1996 |isbn=0-486-68813-5 |url=http://www.gutenberg.org/ebooks/25156}}</ref> (or ''distributive algebra'') over a field ''K'' is a ''K''-vector space ''A'' equipped with a ''K''-[[bilinear map]] <math>A \times A \rightarrow A</math>. The usage of "non-associative" here is meant to convey that associativity is not assumed, but it does not mean it is prohibited – that is, it means "not necessarily associative".
Examples detailed in the main article include:
* [[Euclidean space]] '''R'''<sup>3</sup> with multiplication given by the [[vector cross product]]
* [[Octonion]]s
* [[Lie algebra]]s
* [[Jordan algebra]]s
* [[Alternative algebra]]s
* [[Flexible algebra]]s
* [[Power-associative algebra]]s
==Algebras and rings==
The definition of an associative ''K''-algebra with unit is also frequently given in an alternative way. In this case, an algebra over a field ''K'' is a [[ring (mathematics)|ring]] ''A'' together with a [[ring homomorphism]]
:<math>\eta\colon K\to Z(A),</math>
where ''Z''(''A'') is the [[center (ring theory)|center]] of ''A''. Since ''η'' is a ring homomorphism, then one must have either that ''A'' is the [[zero ring]], or that ''η'' is [[injective function|injective]].  This definition is equivalent to that above, with scalar multiplication
:<math>K\times A \to A</math>
given by
:<math>(k,a) \mapsto \eta(k) a.</math>
Given two such associative unital ''K''-algebras ''A'' and ''B'', a unital ''K''-algebra homomorphism ''f'': ''A'' → ''B'' is a ring homomorphism that commutes with the scalar multiplication defined by ''η'', which one may write as
:<math>f(ka)=kf(a)</math>
for all <math>k\in K</math> and <math>a \in A</math>. In other words, the following diagram commutes:
:<math>\begin{matrix}
&& K &&   \\
& \eta_A \swarrow & \, & \eta_B \searrow & \\
A &&  \begin{matrix} f \\ \longrightarrow \end{matrix}  && B
\end{matrix}</math>
== Structure coefficients ==
{{main|Structure constants}}
For algebras over a field, the bilinear multiplication from ''A'' &times; ''A'' to ''A'' is completely determined by the multiplication of [[basis (linear algebra)|basis]] elements of ''A''.
Conversely, once a basis for ''A'' has been chosen, the products of basis elements can be set arbitrarily, and then extended in a unique way to a bilinear operator on ''A'', i.e., so the resulting multiplication satisfies the algebra laws.
Thus, given the field ''K'', any finite-dimensional algebra can be specified [[up to]] [[isomorphism]] by giving its [[dimension (linear algebra)|dimension]] (say ''n''), and specifying ''n''<sup>3</sup> ''structure coefficients'' ''c''<sub>''i'',''j'',''k''</sub>, which are [[scalar (mathematics)|scalars]].
These structure coefficients determine the multiplication in ''A'' via the following rule:
: <math>\mathbf{e}_{i} \mathbf{e}_{j} = \sum_{k=1}^n c_{i,j,k} \mathbf{e}_{k}</math>
where '''e'''<sub>1</sub>,...,'''e'''<sub>''n''</sub> form a basis of ''A''.
Note however that several different sets of structure coefficients can give rise to isomorphic algebras.
In [[mathematical physics]], the structure coefficients are generally written with upper and lower indices, so as to distinguish their transformation properties under coordinate transformations. Specifically, lower indices are [[covariance and contravariance of vectors|covariant]] indices, and transform via [[pullback (differential geometry)|pullback]]s, while upper indices are [[covariance and contravariance of vectors|contravariant]], transforming under [[Pushforward (differential)|pushforward]]s. Thus, the structure coefficients are often written ''c''<sub>''i'',''j''</sub><sup>''k''</sup>, and their defining rule is written using the [[Einstein notation]] as
: '''e'''<sub>''i''</sub>'''e'''<sub>''j''</sub> = ''c''<sub>''i'',''j''</sub><sup>''k''</sup>'''e'''<sub>''k''</sub>.
If you apply this to vectors written in [[index notation]], then this becomes
: ('''xy''')<sup>''k''</sup> = ''c''<sub>''i'',''j''</sub><sup>''k''</sup>''x''<sup>''i''</sup>''y''<sup>''j''</sup>.
If ''K'' is only a commutative ring and not a field, then the same process works if ''A'' is a [[free module]] over ''K''. If it isn't, then the multiplication is still completely determined by its action on a set that spans ''A''; however, the structure constants can't be specified arbitrarily in this case, and knowing only the structure constants does not specify the algebra up to isomorphism.
== Classification of low-dimensional unital associative algebras over the complex numbers ==
Two-dimensional, three-dimensional and four-dimensional unital associative algebras over the field of complex numbers were completely classified up to isomorphism by [[Eduard Study]].<ref>{{ citation | last=Study | first=E. | year=1890 | title=Über Systeme complexer Zahlen und ihre Anwendungen in der Theorie der Transformationsgruppen | journal=Monatshefte für Mathematik | volume=1 |issue=1 | pages=283–354 | doi=10.1007/BF01692479 | s2cid=121426669 }}</ref>
There exist two such two-dimensional algebras. Each algebra consists of linear combinations (with complex coefficients) of two basis elements, 1 (the identity element) and ''a''. According to the definition of an identity element,
:<math>\textstyle 1 \cdot 1 = 1 \, , \quad 1 \cdot a = a \, , \quad a \cdot 1 = a \, . </math>
It remains to specify
:<math>\textstyle a a = 1 </math> &nbsp; for the first algebra,
:<math>\textstyle a a = 0 </math> &nbsp; for the second algebra.
There exist five such three-dimensional algebras. Each algebra consists of linear combinations of three basis elements, 1 (the identity element), ''a'' and ''b''. Taking into account the definition of an identity element, it is sufficient to specify
:<math>\textstyle a a = a \, , \quad b b = b \, , \quad a b = b a = 0 </math> &nbsp; for the first algebra,
:<math>\textstyle a a = a \, , \quad b b = 0 \, , \quad a b = b a = 0 </math> &nbsp; for the second algebra,
:<math>\textstyle a a = b \, , \quad b b = 0 \, , \quad a b = b a = 0 </math> &nbsp; for the third algebra,
:<math>\textstyle a a = 1 \, , \quad b b = 0 \, , \quad a b = - b a = b </math> &nbsp; for the fourth algebra,
:<math>\textstyle a a = 0 \, , \quad b b = 0 \, , \quad a b = b a = 0 </math> &nbsp; for the fifth algebra.
The fourth of these algebras is non-commutative, and the others are commutative.
== Generalization: algebra over a ring ==
In some areas of mathematics, such as [[commutative algebra]], it is common to consider the more general concept of an '''algebra over a ring''', where a [[commutative ring]] ''R'' replaces the field ''K''. The only part of the definition that changes is that ''A'' is assumed to be an [[Module (mathematics)|''R''-module]] (instead of a ''K''-vector space).
===Associative algebras over rings===
{{main|Associative algebra}}
A [[ring (mathematics)|ring]] ''A'' is always an associative algebra over its [[Center (ring theory)|center]], and over the [[integer]]s. A classical example of an algebra over its center is the [[split-biquaternion|split-biquaternion algebra]], which is isomorphic to <math>\mathbb{H} \times \mathbb{H}</math>, the direct product of two [[quaternion|quaternion algebras]]. The center of that ring is <math>\mathbb{R} \times \mathbb{R}</math>, and hence it has the structure of an algebra over its center, which is not a field. Note that the split-biquaternion algebra is also naturally an 8-dimensional <math>\mathbb{R}</math>-algebra.
In commutative algebra, if ''A'' is a [[commutative ring]], then any unital ring homomorphism <math>R \to A</math> defines an ''R''-module structure on ''A'', and this is what is known as the ''R''-algebra structure.<ref>{{cite book |first=H. |last=Matsumura |title=Commutative Ring Theory |url=https://books.google.com/books?id=yJwNrABugDEC |date=1989 |publisher=Cambridge University Press |isbn=978-0-521-36764-6 |translator-first=M. |translator-last=Reid |edition=2nd |volume=8 |series=Cambridge Studies in Advanced Mathematics }}{{page needed|date=November 2024}}</ref> So a ring comes with a natural <math>\mathbb{Z}</math>-module structure, since one can take the unique homomorphism <math>\mathbb{Z} \to A</math>.<ref>{{cite book |first=Ernst |last=Kunz |title=Introduction to Commutative algebra and algebraic geometry |publisher=Birkhauser |year=1985 |isbn=0-8176-3065-1}}{{page needed|date=November 2024}}</ref> On the other hand, not all rings can be given the structure of an algebra over a field (for example the integers). See ''[[Field with one element]]'' for a description of an attempt to give to every ring a structure that behaves like an algebra over a field.
== See also ==
* [[Algebra over an operad]]
* [[Alternative algebra]]
* [[Clifford algebra]]
* [[Composition algebra]]
* [[Differential algebra]]
* [[Free algebra]]
* [[Geometric algebra]]
* [[Max-plus algebra]]
* [[Mutation (algebra)]]
* [[Operator algebra]]
* [[Zariski's lemma]]
==Notes==
{{reflist}}
==References==
* {{cite book |first1=Michiel |last1=Hazewinkel |author-link=Michiel Hazewinkel |first2=Nadiya |last2=Gubareni |first3=Vladimir V. |last3=Kirichenko |title=Algebras, rings and modules |volume=1 |year=2004 |publisher=Springer |isbn=1-4020-2690-0 }}
[[Category:Algebras| ]]