Algebra over a field

Last updated

In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear". [1]

Contents

The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras 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 numbers 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 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 algebra over a ring. Algebras are not to be confused with vector spaces equipped with a bilinear form, like inner product spaces, 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

Algebravector spacebilinear operator associativity commutativity
complex numbers product of complex numbers
YesYes
cross product of 3D vectors cross product
NoNo (anticommutative)
quaternions Hamilton product
YesNo

Definition

Let K be a field, and let A be a vector space over K equipped with an additional binary operation from A × A to A, denoted here by · (i.e. if x and y are any two elements of A, x · y is the product of x and y). Then A is an algebra over K if the following identities hold for all elements x, y, zA, and all elements (often called scalars) a and b of K:

These three axioms are another way of saying that the binary operation is bilinear. An algebra over K is sometimes also called a K-algebra, and K is called the base field of A. The binary operation is often referred to as multiplication in A. The convention adopted in this article is that multiplication of elements of an algebra is not necessarily 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 K-algebras A and B, a K-algebra homomorphism is a K-linear map f: AB 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 is a bijective 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 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.

  1. x + y is in L (L is closed under addition),
  2. cx is in L (L is closed under scalar multiplication),
  3. 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 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 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 or identity element I with Ix = x = xI for all x in the algebra.

Zero algebra

An algebra is called zero algebra if uv = 0 for all u, v in the algebra, [2] 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 λ, μK and u, vV, then (λ + u) (μ + v) = λμ + (λv + μu). If e1, ... ed is a basis of V, the unital zero algebra is the quotient of the polynomial ring K[E1, ..., En] by the ideal generated by the EiEj for every pair (i, j).

An example of unital zero algebra is the algebra of dual numbers, 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. For example, the theory of Gröbner bases was introduced by Bruno Buchberger for ideals in a polynomial ring R = K[x1, ..., xn] 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

Non-associative algebra

A non-associative algebra [3] (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:

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

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, or that η is injective. 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: AB 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 isomorphism by giving its dimension (say n), and specifying n3structure coefficientsci,j,k, which are scalars. These structure coefficients determine the multiplication in A via the following rule:

where e1,...,en 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 covariant indices, and transform via pullbacks, while upper indices are contravariant, transforming under pushforwards. Thus, the structure coefficients are often written ci,jk, and their defining rule is written using the Einstein notation as

eiej = ci,jkek.

If you apply this to vectors written in index notation, then this becomes

(xy)k = ci,jkxiyj.

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. [4]

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, 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 integers. 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, then any unital ring homomorphism defines an R-module structure on A, and this is what is known as the R-algebra structure. [5] So a ring comes with a natural -module structure, since one can take the unique homomorphism . [6] 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

Notes

  1. See also Hazewinkel, Gubareni & Kirichenko 2004 , p.  3 Proposition 1.1.1
  2. Prolla, João B. (2011) [1977]. "Lemma 4.10". Approximation of Vector Valued Functions. Elsevier. p. 65. ISBN   978-0-08-087136-3.
  3. Schafer, Richard D. (1996). An Introduction to Nonassociative Algebras. ISBN   0-486-68813-5.
  4. Study, E. (1890), "Über Systeme complexer Zahlen und ihre Anwendungen in der Theorie der Transformationsgruppen", Monatshefte für Mathematik, 1 (1): 283–354, doi:10.1007/BF01692479
  5. Matsumura, H. (1989). Commutative Ring Theory. Cambridge Studies in Advanced Mathematics. 8. Translated by Reid, M. (2nd ed.). Cambridge University Press. ISBN   978-0-521-36764-6.
  6. Kunz, Ernst (1985). Introduction to Commutative algebra and algebraic geometry. Birkhauser. ISBN   0-8176-3065-1.

Related Research Articles

Associative algebra Algebraic structure with (a + b)(c + d) = ac + ad + bc + bd and (a)(bc) = (ab)(c)

In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication, and a scalar multiplication by elements in some field. 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 K-algebra to mean an associative algebra over the field K. A standard first example of a K-algebra is a ring of square matrices over a field K, with the usual matrix multiplication.

In algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a nonzero ring in which every nonzero element a has a multiplicative inverse, that is, an element generally denoted a–1, such that aa–1 = a–1a = 1. So, division may be defined as a / b = ab–1, but this notation is generally avoided, as one may have ab–1b–1a.

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type. The word homomorphism comes from the Ancient Greek language: ὁμός meaning "same" and μορφή meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).

Integral domain Commutative ring with no zero divisors other than zero

In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c.

Lie algebra Vector space with a binary operation satisfying the Jacobi identity

In mathematics, a Lie algebra is a vector space together with an operation called the Lie bracket, an alternating bilinear map , that satisfies the Jacobi identity. The vector space together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative.

Ring (mathematics) Algebraic structure with addition and multiplication

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

Quaternion Noncommutative extension of the real numbers

In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative.

In abstract algebra, the endomorphisms of an abelian group X form a ring. This ring is called the endomorphism ringX, denoted by End(X); the set of all homomorphisms of X into itself. Addition of endomorphisms arises naturally in a pointwise manner and multiplication via endomorphism composition. Using these operations, the set of endomorphisms of an abelian group forms a (unital) ring, with the zero map as additive identity and the identity map as multiplicative identity.

Commutative ring Algebraic structure

In ring theory, a branch of abstract algebra, 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 noncommutative rings where multiplication is not required to be commutative.

In mathematics, an algebraic structure consists of a nonempty set A, a collection of operations on A of finite arity, and a finite set of identities, known as axioms, that these operations must satisfy.

Exterior algebra Algebraic construction used in multilinear algebra and geometry

In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors and , denoted by , is called a bivector and lives in a space called the exterior square, a vector space that is distinct from the original space of vectors. The magnitude of can be interpreted as the area of the parallelogram with sides and , which in three dimensions can also be computed using the cross product of the two vectors. More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like the cross product, the exterior product is anticommutative, meaning that for all vectors and , but, unlike the cross product, the exterior product is associative.

Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject.

Module (mathematics) Generalization of vector space, with scalars in a ring instead of a field

In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra. A module over a ring is a generalization of the notion of vector space over a field, wherein the corresponding scalars are the elements of an arbitrary given ring and a multiplication is defined between elements of the ring and elements of the module. A module taking its scalars from a ring R is called an R-module.

In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is one-to-one with the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring.

Polynomial ring Algebraic structure

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates with coefficients in another ring, often a field.

In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.

In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a ring. If the algebra is not unital, it may be made so in a standard way ; there is no essential difference between modules for the resulting unital ring, in which the identity acts by the identity mapping, and representations of the algebra.

In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality theories. Frobenius algebras began to be studied in the 1930s by Richard Brauer and Cecil Nesbitt and were named after Ferdinand Frobenius. Tadashi Nakayama discovered the beginnings of a rich duality theory, . Jean Dieudonné used this to characterize Frobenius algebras. Frobenius algebras were generalized to quasi-Frobenius rings, those Noetherian rings whose right regular representation is injective. In recent times, interest has been renewed in Frobenius algebras due to connections to topological quantum field theory.

In mathematics, and more specifically in abstract algebra, a rng is an algebraic structure satisfying the same properties as a ring, but without assuming the existence of a multiplicative identity. The term "rng" is meant to suggest that it is a "ring" without "i", that is, without the requirement for an "identity element".

Non-associative algebra Algebra over a field where binary multiplication is not necessarily associative

A non-associative algebra is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure A is a non-associative algebra over a field K if it is a vector space over K and is equipped with a K-bilinear binary multiplication operation A × AA which may or may not be associative. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped with the cross product operation. Since it is not assumed that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions (ab)(cd), d and a(b ) may all yield different answers.

References