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Encyclopedia > Division ring

In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i.e. an element x with ax = xa = 1). If rings are viewed as categorical constructions, then this is equivalent to requiring that all nonzero morphisms are isomorphisms. Division rings are very similar to fields - they differ only in that their multiplication is not required to be commutative. The condition 0 ≠ 1 is only there to exclude the trivial ring with a single element 0 = 1. Stated differently, a ring is a division ring iff the group of units is the set of all non-zero elements. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... The reciprocal function: y = 1/x. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... It has been suggested that this article or section be merged with Logical biconditional. ... In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of...

All fields are division rings; more interesting examples are the non-commutative division rings. The best known example is the ring of quaternions 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 the endomorphism ring of S is a division ring; every division ring arises in this fashion from some simple module. In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, the real numbers may be described informally in several different ways. ... In abstract algebra, a (left or right) module S over a ring R is called simple if it is not the zero module and if its only submodules are 0 and S. Understanding the simple modules over a ring is usually helpful because they form the building blocks of all... In abstract algebra, one associates to certain objects a ring, the objects endomorphism ring, which encodes several internal properties of the object. ...

Much of linear algebra may be formulated, and remains correct, for modules over division rings instead of vector spaces 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. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. ... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In mathematics, Gaussian elimination (not to be confused with Gaussâ€“Jordan elimination), named after Carl Friedrich Gauss, is an algorithm in linear algebra for determining the solutions of a system of linear equations, for determining the rank of a matrix, and for calculating the inverse of an invertible square matrix. ...

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. The term center is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. ... In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division is possible. ...

Wedderburn's (little) theorem: All finite division rings are commutative and therefore finite fields. (Ernst Witt gave a simple proof.) Joseph Henry Maclagen Wedderburn (2 February 1882- 9 October 1948) was a Scottish mathematician, who from 1909 had positions at Princeton University. ... In abstract algebra, a finite field or Galois field (so named in honor of Ã‰variste Galois) is a field that contains only finitely many elements. ... Ernst Witt (June 26, 1911-July 3, 1991) was a German mathematician born in Als. ...

Frobenius theorem: The only finite dimensional division algebra over the reals are the real numbers, the complex numbers and the quaternions. 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. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...

Division rings used to be called fields in an older usage, which remained in other languages. A more complete comparison is found in the article Field (mathematics). In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

While division rings and algebras as discussed here are assumed to have associative multiplication, nonassociative division algebras such as the octonions are also of interest. In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division is possible. ... In mathematics, the octonions are a nonassociative extension of the quaternions. ...

Results from FactBites:

 Division ring - definition of Division ring in Encyclopedia (243 words) In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i.e. Division rings are very similar to fields except that their multiplication is not required to be commutative. Wedderburn's theorem states that all finite division rings are commutative and therefore finite fields.
 Division ring - Wikipedia, the free encyclopedia (445 words) Stated differently, a ring is a division ring iff the group of units is the set of all non-zero elements. The center of a division ring is commutative and therefore a field. Division rings can be roughly classified according to whether or not they are finite-dimensional or infinite-dimensional over their centers.
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