In algebra, a division ring, also called a skew field (or, occasionally, a sfield[1][2]), is a nontrivial ring in which division by nonzero elements is defined.
[7] 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).
If one allows only rational instead of real coefficients in the constructions of the quaternions, one obtains another division ring.
Much of linear algebra may be formulated, and remains correct, for modules over a division ring D instead of vector spaces over a field.
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.
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 ring of Hamiltonian quaternions forms a four-dimensional algebra over its center, which is isomorphic to the real numbers.
Division rings used to be called "fields" in an older usage.
A near-field is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws.