Division ring
Encyclopedia
In abstract algebra
, a division ring, also called a skew field, is a ring
in which division
is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse
, i.e., an element x with . Stated differently, a ring is a division ring if and only if the group of units is the set of all non-zero elements.
Division rings differ from fields
only in that their multiplication is not required to be commutative. However, by Wedderburn's little theorem
all finite division rings are commutative and therefore finite field
s. Historically, division rings were sometimes referred to as fields, while fields were called “commutative fields”.
s H. If we allow only rational
instead of real
coefficients in the constructions of the quaternions, we obtain another division ring. In general, if R is a ring and S is a simple module
over R, then, by the Schur's lemma
, the endomorphism ring
of S is a division ring; every division ring arises in this fashion from some simple module.
Much of linear algebra
may be formulated, and remains correct, for (left) modules
over division rings instead of vector space
s over fields. Every module over a division ring has a basis; linear maps between finite-dimensional modules over a division ring can be described by matrices
, and the Gaussian elimination
algorithm remains applicable. Differences between linear algebra over fields and skew fields occur whenever the order of the factors in a product matters. For example, the proof that the column rank of a matrix over a field equals its row rank yields for matrices over division rings only that the left column rank equals its right row rank: it does not make sense to speak about the rank of a matrix over a division ring.
The center of a division ring is commutative and therefore a field. Every division ring is therefore a division algebra
over its center. Division rings can be roughly classified according to whether or not they are finite-dimensional or infinite-dimensional over their centers. The former are called centrally finite and the latter centrally infinite. Every field is, of course, one-dimensional over its center. The quaternion ring forms a 4-dimensional algebra over its center, which is isomorphic to the real numbers.
: All finite division rings are commutative and therefore finite field
s. (Ernst Witt
gave a simple proof.)
Frobenius theorem
: The only finite-dimensional division algebras over the reals are the reals themselves, the complex number
s, and the quaternion
s.
.
Skew fields have an interesting semantic
feature: a modifier (here "skew") widens the scope of the base term (here "field"). Thus a field is a particular type of skew field, and not all skew fields are fields.
While division rings and algebras as discussed here are assumed to have associative multiplication, nonassociative division algebras such as the octonion
s are also of interest.
A near-field
is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, a division ring, also called a skew field, is a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
in which division
Division (mathematics)
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse
Multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...
, i.e., an element x with . Stated differently, a ring is a division ring if and only if the group of units is the set of all non-zero elements.
Division rings differ from fields
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
only in that their multiplication is not required to be commutative. However, by Wedderburn's little theorem
Wedderburn's little theorem
In mathematics, Wedderburn's little theorem states that every finite domain is a field. In other words, for finite rings, there is no distinction between domains, skew-fields and fields.The Artin–Zorn theorem generalizes the theorem to alternative rings....
all finite division rings are commutative and therefore finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
s. Historically, division rings were sometimes referred to as fields, while fields were called “commutative fields”.
Relation to fields and linear algebra
All fields are division rings; more interesting examples are the non-commutative division rings. The best known example is the ring of quaternionQuaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
s H. If we allow only rational
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
instead of real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
coefficients in the constructions of the quaternions, we obtain another division ring. In general, if R is a ring and S is a simple module
Simple module
In mathematics, specifically in ring theory, the simple modules over a ring R are the modules over R which have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M...
over R, then, by the Schur's lemma
Schur's lemma
In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if M and N are two finite-dimensional irreducible representations...
, the endomorphism ring
Endomorphism ring
In abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object; this may be denoted End...
of S is a division ring; every division ring arises in this fashion from some simple module.
Much of linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
may be formulated, and remains correct, for (left) modules
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
over division rings instead of vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s over fields. Every module over a division ring has a basis; linear maps between finite-dimensional modules over a division ring can be described by matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
, and the Gaussian elimination
Gaussian elimination
In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix...
algorithm remains applicable. Differences between linear algebra over fields and skew fields occur whenever the order of the factors in a product matters. For example, the proof that the column rank of a matrix over a field equals its row rank yields for matrices over division rings only that the left column rank equals its right row rank: it does not make sense to speak about the rank of a matrix over a division ring.
The center of a division ring is commutative and therefore a field. Every division ring is therefore a division algebra
Division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field, in which division is possible.- Definitions :...
over its center. Division rings can be roughly classified according to whether or not they are finite-dimensional or infinite-dimensional over their centers. The former are called centrally finite and the latter centrally infinite. Every field is, of course, one-dimensional over its center. The quaternion ring forms a 4-dimensional algebra over its center, which is isomorphic to the real numbers.
Ring theorems
Wedderburn's little theoremWedderburn's little theorem
In mathematics, Wedderburn's little theorem states that every finite domain is a field. In other words, for finite rings, there is no distinction between domains, skew-fields and fields.The Artin–Zorn theorem generalizes the theorem to alternative rings....
: All finite division rings are commutative and therefore finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
s. (Ernst Witt
Ernst Witt
Ernst Witt was a German mathematician born on the island of Als . Shortly after his birth, he and his parents moved to China, and he did not return to Europe until he was nine....
gave a simple proof.)
Frobenius theorem
Frobenius theorem (real division algebras)
In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers...
: The only finite-dimensional division algebras over the reals are the reals themselves, the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s, and the quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
s.
Related notions
Division rings used to be called "fields" in an older usage. In many languages, a word meaning "body" is used for division rings, in some languages designating either commutative or non-commutative division rings, while in others specifically designating commutative division rings (what we now call fields in English). A more complete comparison is found in the article Field (mathematics)Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
.
Skew fields have an interesting semantic
Lexical semantics
Lexical semantics is a subfield of linguistic semantics. It is the study of how and what the words of a language denote . Words may either be taken to denote things in the world, or concepts, depending on the particular approach to lexical semantics.The units of meaning in lexical semantics are...
feature: a modifier (here "skew") widens the scope of the base term (here "field"). Thus a field is a particular type of skew field, and not all skew fields are fields.
While division rings and algebras as discussed here are assumed to have associative multiplication, nonassociative division algebras such as the octonion
Octonion
In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H...
s are also of interest.
A near-field
Near-field (mathematics)
In mathematics, a near-field is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws. Alternatively, a near-field is a near-ring in which there is a multiplicative identity, and every non-zero element has a multiplicative inverse.- Definition...
is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws.