Division ring

Division ring

A division ring is a type of ring where every non-zero element has a multiplicative inverse, meaning that every non-zero element is a unit. In other words, the group of units in a division ring is equal to the set of all non-zero elements. An example of a division ring that is not a field is the set of quaternions, which is represented by a + bi + cj + dk, where a, b, c, and d are real numbers, i, j, and k are pure imaginary, and certain algebraic laws are maintained.From: CRC Standard Mathematical Tables and Formulas [2018], General remainder theorem and factor theorem for polynomials over non-commutative coefficient rings [2020], College Geometry [2019]

Preliminaries

View Chapter

Purchase Book

Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017

J. Tinsley Oden, Leszek F. Demkowicz

Note that a field is also a commutative division ring (though not every division ring is a field). The sets of real, rational, and complex numbers individually form fields, as does the set of all square diagonal matrices of a specified order.

General remainder theorem and factor theorem for polynomials over non-commutative coefficient rings

View Article

Journal Information

Published in International Journal of Mathematical Education in Science and Technology, 2020

A. Cuida, F. Laudano, E. Martinez-Moro

In abstract algebra, a division ring (cf. Lam & Leroy, 1988; Martínez-Penas, 2018), also called a skew field (cf. Smits, 1968), is a ring in which division is possible. Specifically, it is a non-zero ring in which every non-zero element a has a multiplicative inverse, i.e. an element x with ax = xa = 1. Stated differently, a ring is a division ring if and only if the group of units equals the set of all non-zero elements. A division ring is a type of non-commutative ring.

Explore chapters and articles related to this topic

Preliminaries

General remainder theorem and factor theorem for polynomials over non-commutative coefficient rings