Editing Division ring (section)

{{Short description|Algebraic structure also called skew field}} 
In [[algebra]], a '''division ring''', also called a '''skew field''' (or, occasionally, a '''sfield'''<ref>{{Cite web |title=Definition:Skew Field - ProofWiki |url=https://proofwiki.org/wiki/Definition:Skew_Field |access-date=2024-10-13 |website=proofwiki.org}}</ref><ref>{{Cite journal |last=Hua |first=Loo-Keng |date=1949 |title=Some Properties of a Sfield |journal=Proceedings of the National Academy of Sciences |language=en |volume=35 |issue=9 |pages=533–537 |doi=10.1073/pnas.35.9.533 |doi-access=free |issn=0027-8424 |pmc=1063075 |pmid=16588934|bibcode=1949PNAS...35..533H }}</ref>), is a [[zero ring|nontrivial]] [[ring (mathematics)|ring]] in which [[division (mathematics)|division]] by nonzero elements is defined. Specifically, it is a nontrivial ring{{refn|In this article, rings have a {{math|1}}.}} in which every nonzero element {{mvar|a}} has a [[multiplicative inverse]], that is, an element usually denoted {{math|''a''{{sup|–1}}}}, such that {{math|1=''a{{space|thin}}a''{{sup|–1}} = ''a''{{sup|–1}}{{space|thin}}''a'' = 1}}. So, (right) ''division'' may be defined as {{math|1=''a'' / ''b'' = ''a''{{space|thin}}''b''<sup>–1</sup>}}, but this notation is avoided, as one may have {{math|1=''a{{space|thin}}b''{{sup|–1}} ≠ ''b''{{sup|–1}}{{space|thin}}''a''}}.

A commutative division ring is a [[Field (mathematics)|field]]. [[Wedderburn's little theorem]] asserts that all finite division rings are commutative and therefore [[finite field]]s. 

Historically, division rings were sometimes referred to as fields, while fields were called "commutative fields".{{refn|Within the English language area the terms "skew field" and "sfield" were mentioned 1948 by Neal McCoy<ref>1948, Rings and Ideals. Northampton, Mass., Mathematical Association of America</ref> as "sometimes used in the literature", and since 1965 ''skewfield'' has an entry in the [[OED]]. The German term {{ill|Schiefkörper|de|vertical-align=sup}} is documented, as a suggestion by [[van der Waerden]], in a 1927 text by [[Emil Artin]],{{refn|{{citation|last1=Artin|first1=Emil|year=1965|title=Collected Papers|editor1=Serge Lang|editor2=John T. Tate|location=New York|publisher=Springer}}}} and was used by [[Emmy Noether]] as lecture title in 1928.<ref>{{citation|last1=Brauer|first1=Richard|year=1932|title=Über die algebraische Struktur von Schiefkörpern|journal=Journal für die reine und angewandte Mathematik|volume=166 |issue=4|pages=103–252}}</ref>}} In some languages, such as [[French language|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 ring|simple]]. That is, they have no two-sided [[Ideal (ring theory)|ideal]] besides the [[zero ideal]] and itself.

{{Algebraic structures |Ring}}