In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e., an element x with a·x = x·a = 1. Stated differently, a ring is a division ring if and only if the group of units is the set of all non-zero elements.
Division rings differ from fields only in that their multiplication is not required to be commutative. However, by Wedderburn's little theorem 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”.
Read more about Division Ring: Relation To Fields and Linear Algebra, Examples, Ring Theorems, Related Notions
Famous quotes containing the words division and/or ring:
“Major [William] McKinley visited me. He is on a stumping tour.... I criticized the bloody-shirt course of the canvass. It seems to me to be bad “politics,” and of no use.... It is a stale issue. An increasing number of people are interested in good relations with the South.... Two ways are open to succeed in the South: 1. A division of the white voters. 2. Education of the ignorant. Bloody-shirt utterances prevent division.”
—Rutherford Birchard Hayes (1822–1893)
“What is a novel? I say: an invented story. At the same time a story which, though invented has the power to ring true. True to what? True to life as the reader knows life to be or, it may be, feels life to be. And I mean the adult, the grown-up reader. Such a reader has outgrown fairy tales, and we do not want the fantastic and the impossible. So I say to you that a novel must stand up to the adult tests of reality.”
—Elizabeth Bowen (1899–1973)