In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an algebra over a field (often simply called an algebra) is a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
equipped with a
bilinear product. Thus, an algebra is an
algebraic structure
In mathematics, an algebraic structure 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 multiplication), and a finite set o ...
consisting of a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
together with operations of multiplication and addition 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 b ...
by elements of a
field and satisfying the axioms implied by "vector space" and "bilinear".
The multiplication operation in an algebra may or may not be
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
, leading to the notions of
associative algebra
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 multiplic ...
s and
non-associative algebras. Given an integer ''n'', the
ring of
real square matrices of order ''n'' is an example of an associative algebra over the field of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s under
matrix addition and
matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
since matrix multiplication is associative. Three-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
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
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
with respect to the multiplication. The ring of real square matrices of order ''n'' forms a unital algebra since the
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...
of order ''n'' is the identity element with respect to matrix multiplication. It is an example of a unital associative algebra, a
(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
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, ''unital associative commutative algebra''.
Replacing the field of scalars by a
commutative ring
In mathematics, a commutative ring is a 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 properties that are not ...
leads to the more general notion of an
algebra over a ring. Algebras are not to be confused with vector spaces equipped with a
bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
, like
inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
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
Definition
Let be a field, and let be a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over equipped with an additional
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
from to , denoted here by (that is, if and are any two elements of , then is an element of that is called the ''product'' of and ). Then is an ''algebra'' over if the following identities hold for all elements in , and all elements (often called
scalars) and in :
* Right
distributivity
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmeti ...
:
* Left distributivity:
* Compatibility with scalars: .
These three axioms are another way of saying that the binary operation is
bilinear. An algebra over is sometimes also called a ''-algebra'', and is called the ''base field'' of . The binary operation is often referred to as ''multiplication'' in . The convention adopted in this article is that multiplication of elements of an algebra is not necessarily
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
, although some authors use the term ''algebra'' to refer to an
associative algebra
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 multiplic ...
.
When a binary operation on a vector space 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. Most familiar as the name of ...
, 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 ''K''-algebras ''A'' and ''B'', a ''K''-algebra
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 ...
is a ''K''-
linear map
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 ...
''f'': ''A'' → ''B'' such that ''f''(xy) = ''f''(x) ''f''(y) for all x, y in ''A''. The space of all ''K''-algebra homomorphisms between ''A'' and ''B'' is frequently written as
:
A ''K''-algebra
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
is a
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
''K''-algebra homomorphism. For all practical purposes, isomorphic algebras differ only by notation.
Subalgebras and ideals
A ''subalgebra'' of an algebra over a field ''K'' is a
linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...
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. Notice that 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.
It is important to notice that this definition is different from the definition of an
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
If we have a
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
''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
. So if ''A'' is an algebra over ''K'', then
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
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. Most familiar as the name of ...
or
associativity
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
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
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (a ...
or identity element ''I'' with ''Ix'' = ''x'' = ''xI'' for all ''x'' in the algebra.
Zero algebra
An algebra is called zero algebra if for all ''u'', ''v'' in the algebra, 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 and , then . If is a basis of ''V'', the unital zero algebra is the quotient of the polynomial ring by the
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
generated by the ''E''
''i''''E''
''j'' for every pair .
An example of unital zero algebra is the algebra of
dual number
In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0.
Du ...
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
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
. For example, the theory of
Gröbner bases was introduced by
Bruno Buchberger for
ideals in a polynomial ring 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
Examples of associative algebras include
* the algebra of all ''n''-by-''n''
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
over a field (or commutative ring) ''K''. Here the multiplication is ordinary
matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
.
*
group algebras, where a
group serves as a basis of the vector space and algebra multiplication extends group multiplication.
* the commutative algebra ''K''
'x''of all
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s over ''K'' (see
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
).
* algebras of
functions, such as the R-algebra of all real-valued
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
functions defined on the
interval ,1 or the C-algebra of all
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...
s defined on some fixed open set in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
. These are also commutative.
*
Incidence algebras are built on certain
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
s.
* algebras of
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 Map (mathematics), mapping V \to W between two vect ...
s, for example on a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. Here the algebra multiplication is given by the
composition of operators. These algebras also carry a
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
; many of them are defined on an underlying
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) 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 vector ...
, which turns them into
Banach algebras. If an involution is given as well, we obtain
B*-algebras and
C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuou ...
s. These are studied in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
.
Non-associative algebra
A ''non-associative algebra''
(or ''distributive algebra'') over a field ''K'' is a ''K''-vector space ''A'' equipped with a ''K''-
bilinear map . 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
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
R
3 with multiplication given by the
vector cross product
*
Octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...
s
*
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 identi ...
s
*
Jordan algebras
*
Alternative algebras
*
Flexible algebras
*
Power-associative algebras
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 ''A'' together with a
ring homomorphism
In ring theory, a branch of abstract algebra, 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 such that ''f'' is:
:addition preser ...
:
where ''Z''(''A'') is the
center of ''A''. Since ''η'' is a ring homomorphism, then one must have either that ''A'' is the
zero ring
In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which for ...
, or that ''η'' is
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
. This definition is equivalent to that above, with scalar multiplication
:
given by
:
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
:
for all
and
. In other words, the following diagram commutes:
:
Structure coefficients
For algebras over a field, the bilinear multiplication from ''A'' × ''A'' to ''A'' is completely determined by the multiplication of
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 Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' with respect to ''R'' ...
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
by giving its
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
(say ''n''), and specifying ''n''
3 ''structure coefficients'' ''c''
''i'',''j'',''k'', which are
scalars
Scalar may refer to:
* Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
.
These structure coefficients determine the multiplication in ''A'' via the following rule:
:
where e
1,...,e
''n'' form a basis of ''A''.
Note however that several different sets of structure coefficients can give rise to isomorphic algebras.
In
mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developm ...
, 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
covariant indices, and transform via
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: ...
s, while upper indices are
contravariant, transforming under
pushforwards. Thus, the structure coefficients are often written ''c''
''i'',''j''''k'', and their defining rule is written using the
Einstein notation
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
as
: e
''i''e
''j'' = ''c''
''i'',''j''''k''e
''k''.
If you apply this to vectors written in
index notation
In mathematics and computer programming, index notation is used to specify the elements of an array of numbers. The formalism of how indices are used varies according to the subject. In particular, there are different methods for referring to t ...
, then this becomes
: (xy)
''k'' = ''c''
''i'',''j''''k''''x''
''i''''y''
''j''.
If ''K'' is only a commutative ring and not a field, then the same process works if ''A'' is a
free module
In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a fiel ...
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.
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,
:
It remains to specify
:
for the first algebra,
:
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
:
for the first algebra,
:
for the second algebra,
:
for the third algebra,
:
for the fourth algebra,
:
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
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
, it is common to consider the more general concept of an algebra over a ring, where a commutative unital ring ''R'' replaces the field ''K''. The only part of the definition that changes is that ''A'' is assumed to be an
''R''-module (instead of a vector space over ''K'').
Associative algebras over rings
A
ring ''A'' is always an associative algebra over its
center, and over the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s. A classical example of an algebra over its center is the
split-biquaternion algebra, which is isomorphic to
, the direct product of two
quaternion algebras. The center of that ring is
, 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
-algebra.
In commutative algebra, if ''A'' is a
commutative ring
In mathematics, a commutative ring is a 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 properties that are not ...
, then any unital ring homomorphism
defines an ''R''-module structure on ''A'', and this is what is known as the ''R''-algebra structure. So a ring comes with a natural
-module structure, since one can take the unique homomorphism
.
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
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperco ...
*
Differential algebra
*
Free algebra
*
Geometric algebra
In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ...
*
Max-plus algebra
*
Mutation (algebra)
*
Operator algebra
*
Zariski's lemma In algebra, Zariski's lemma, proved by , states that, if a field is finitely generated as an associative algebra over another field , then is a finite field extension of (that is, it is also finitely generated as a vector space).
An important ...
Notes
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