extension by localization
Let be a commutative ring and let be a non-empty multiplicative subset of . Then the localisation (http://planetmath.org/Localization) of at gives the commutative ring but, generally, it has no subring isomorphic to . Formally, consists of all elements (, ). Therefore, is called also a ring of quotients of . If , then ; we assume now that .
If, especially, contains no zero divisors of the ring , then the above mapping is an isomorphism from to a certain subring of , and we may think that . In this case, the ring of fractions of is an extension ring of ; this concerns of course the case that is an integral domain. But if is a finite ring, then , and no proper extension is obtained.
|Title||extension by localization|
|Date of creation||2013-03-22 14:24:42|
|Last modified on||2013-03-22 14:24:42|
|Last modified by||pahio (2872)|
|Synonym||ring extension by localization|
|Defines||ring of fractions|
|Defines||ring of quotients|