In

^{op} in order for the rule to remain valid.
Every module over a division ring is free; that is, it has a basis, and all bases of a module have the same number of elements. Linear maps between finite-dimensional modules over a division ring can be described by matrices; the fact that linear maps by definition commute with scalar multiplication is most conveniently represented in notation by writing them on the ''opposite'' side of vectors as scalars are. The

Proof of Wedderburn's Theorem at Planet Math

Grillet's Abstract Algebra, section VIII.5's characterization of division rings via their free modules.

Ring theory

algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...

, a division ring, also called a skew field, is a nontrivial ring in which division
Division or divider may refer to:
Mathematics
*Division (mathematics), the inverse of multiplication
* Division algorithm, a method for computing the result of mathematical division
Military
*Division (military), a formation typically consisting ...

by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a 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 ...

, that is, an element usually denoted , such that . So, (right) ''division'' may be defined as , but this notation is avoided, as one may have .
A commutative division ring is a field. 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, division rings and fields.
The Artin–Zorn theorem generalizes the theorem to a ...

asserts that all finite division rings are commutative and therefore finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...

s.
Historically, division rings were sometimes referred to as fields, while fields were called "commutative fields". In some languages, such as French, the word equivalent to "field" ("corps") is used for both commutative and noncommutative cases, and the distinction between the two cases is made by adding qualificatives such as "corps commutatif" (commutative field) or "corps gauche" (skew field).
All division rings are simple. That is, they have no two-sided ideal besides the zero ideal and itself.
Relation to fields and linear algebra

All fields are division rings; more interesting examples are the noncommutative division rings. The best known example is the ring ofquaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...

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 In mathematics, specifically in ring theory, the simple modules over a ring ''R'' are the (left or right) modules over ''R'' that are non-zero and have no non-zero proper submodules. Equivalently, a module ''M'' is simple if and only if every ...

over ''R'', then, by 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
of a group ...

, 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 modules over a division ring ''D'' instead of 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 ...

s over a field. Doing so it must be specified whether one is considering right or left modules, and some care is needed in properly distinguishing left and right in formulas. Working in coordinates, elements of a finite dimensional right module can be represented by column vectors, which can be multiplied on the right by scalars, and on the left by matrices (representing linear maps); for elements of a finite dimensional left module, row vectors must be used, which can be multiplied on the left by scalars, and on the right by matrices. The dual of a right module is a left module, and vice versa. The transpose of a matrix must be viewed as a matrix over the opposite division ring ''D''Gaussian elimination
In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...

algorithm remains applicable. The column rank of a matrix is the dimension of the right module generated by the columns, and the row rank is dimension of the left module generated by the rows; the same proof as for the vector space case can be used to show that these ranks are the same, and define the rank of a matrix.
In fact the converse is also true and this gives a ''characterization of division rings'' via their module category: A unital ring ''R'' is a division ring if and only if every ''R''- module is free.
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 ring of Hamiltonian quaternions forms a 4-dimensional algebra over its center, which is isomorphic to the real numbers.
Examples

* As noted above, allfields
Fields may refer to:
Music
*Fields (band), an indie rock band formed in 2006
*Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song by ...

are division rings.
* The quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...

s form a noncommutative division ring.
* The subset of the quaternions , such that , , , and belong to a fixed subfield of the 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, is a noncommutative division ring. When this subfield is the field of rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rati ...

s, this is the division ring of ''rational quaternions''.
* Let $\backslash sigma:\; \backslash Complex\; \backslash to\; \backslash Complex$ be an automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphism ...

of the field Let $\backslash Complex((z,\backslash sigma))$ denote the ring of formal Laurent series with complex coefficients, wherein multiplication is defined as follows: instead of simply allowing coefficients to commute directly with the indeterminate for define $z^i\backslash alpha\; :=\; \backslash sigma^i(\backslash alpha)\; z^i$ for each index If $\backslash sigma$ is a non-trivial automorphism of complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

s (such as the conjugation), then the resulting ring of Laurent series is a noncommutative division ring known as a ''skew Laurent series ring'';Lam (2001), p. 10 if then it features the standard multiplication of formal series. This concept can be generalized to the ring of Laurent series over any fixed field given a nontrivial
Main theorems

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, division rings and fields.
The Artin–Zorn theorem generalizes the theorem to a ...

: All finite division rings are commutative and therefore finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...

s. ( Ernst Witt gave a simple proof.)
Frobenius theorem: The only finite-dimensional associative division algebras over the reals are the reals themselves, the complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

s, and the quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...

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 noncommutative 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 onfields
Fields may refer to:
Music
*Fields (band), an indie rock band formed in 2006
*Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song by ...

.
The name "Skew field" has an interesting semantic
Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and compu ...

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
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 have ...

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 law
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 arithmetic, ...

s.
Notes

See also

* Hua's identityReferences

*Further reading

* {{cite book , last=Cohn , first=P.M. , author-link=Paul Cohn , title=Skew fields. Theory of general division rings , zbl=0840.16001 , series=Encyclopedia of Mathematics and Its Applications , volume=57 , location=Cambridge , publisher=Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer.
Cambridge University Pres ...

, year=1995 , isbn=0-521-43217-0 , url-access=registration , url=https://archive.org/details/skewfieldstheory0000cohn
External links

Proof of Wedderburn's Theorem at Planet Math

Grillet's Abstract Algebra, section VIII.5's characterization of division rings via their free modules.

Ring theory