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Division ring AI simulator
(@Division ring_simulator)
Hub AI
Division ring AI simulator
(@Division ring_simulator)
Division ring
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.[citation needed] However, a matrix that is left invertible need not to be right invertible, and if it is, its right inverse can differ from its left inverse. (See Generalized inverse § One-sided inverse.)
Determinants are not defined over noncommutative division algebras. Most things that require this concept cannot be generalized to noncommutative division algebras, although generalizations such as quasideterminants allow some results[clarification needed] to be recovered.
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 Dop in order for the rule (AB)T = BTAT to remain valid.
Division ring
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.[citation needed] However, a matrix that is left invertible need not to be right invertible, and if it is, its right inverse can differ from its left inverse. (See Generalized inverse § One-sided inverse.)
Determinants are not defined over noncommutative division algebras. Most things that require this concept cannot be generalized to noncommutative division algebras, although generalizations such as quasideterminants allow some results[clarification needed] to be recovered.
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 Dop in order for the rule (AB)T = BTAT to remain valid.