Cartan–Brauer–Hua theorem
{{short description|Result pertaining to division rings}}
In abstract algebra, the Cartan–Brauer–Hua theorem (named after Richard Brauer, Élie Cartan, and Hua Luogeng) is a theorem pertaining to division rings. It says that given two division rings {{nowrap|K ⊆ D}} such that xKx−1 is contained in K for every x not equal to 0 in D, either K is contained in the center of D, or {{nowrap|1=K = D}}. In other words, if the unit group of K is a normal subgroup of the unit group of D, then either {{nowrap|1=K = D}} or K is central {{harv|Lam|2001|p=211}}.
References
{{reflist}}
- {{cite book |last=Herstein |first=I. N. |author-link=Israel Nathan Herstein |title=Topics in algebra |publisher=Wiley |location=New York |year=1975 |page=[https://archive.org/details/topicsinalgebra00hers/page/368 368] |isbn=0-471-01090-1 |url-access=registration |url=https://archive.org/details/topicsinalgebra00hers/page/368 }}
- {{Cite book | last1=Lam | first1=Tsit-Yuen | title=A First Course in Noncommutative Rings | publisher=Springer-Verlag | location=Berlin, New York | edition=2nd | isbn=978-0-387-95325-0 | mr=1838439 | year=2001 }}
{{DEFAULTSORT:Cartan-Brauer-Hua theorem}}
Category:Theorems in ring theory
{{Abstract-algebra-stub}}