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Concept
Division ring
Summary
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element a has a multiplicative inverse, that is, an element usually denoted a^–1, such that a a^–1 = a^–1 a = 1. So, (right) division may be defined as a / b = a b–1, but this notation is avoided, as one may have a b^–1 ≠ b^–1 a.
A commutative division ring is a field. Wedderburn's little theorem asserts that all finite division rings are commutative and therefore finite fields.
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.
All fields are division rings, and every non-field division ring is noncommutative. The best known example is the ring of quaternions. If one allows only rational instead of real coefficients in the constructions of the quaternions, one obtains another division ring. In general, if R is a ring and S is a simple module over R, then, by 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 modules over a division ring D instead of vector spaces over a field. Doing so, one must specify whether one is considering right or left modules, and some care is needed in properly distinguishing left and right in formulas. In particular, every module has a basis, and Gaussian elimination can be used. So, everything that can be defined with these tools works on division algebras. Matrices and their products are defined similarly.
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Related courses (1)
MATH-334: Representation theory
Study the basics of representation theory of groups and associative algebras.
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Related concepts (24)
Ring (mathematics)
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.
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers.
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 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. Quaternions are generally represented in the form where a, b, c, and d are real numbers; and 1, i, j, and k are the basis vectors or basis elements.