finite ring has no proper overrings
The regular elements of a finite commutative ring are the units of the ring (see the parent (http://planetmath.org/NonZeroDivisorsOfFiniteRing) of this entry). Generally, the largest overring of , the total ring of fractions , is obtained by forming , the extension by localization, using as the multiplicative set the set of all regular elements, which in this case is the unit group of . The ring may be considered as a subring of , which consists formally of the fractions with and . Since every has its own group inverse in and so in , it’s evident that no other elements than the elements of . Consequently, , and therefore also any overring of coincides with .
Accordingly, one can not extend a finite commutative ring by using a localization. Possible extensions must be made via some kind of adjunction (http://planetmath.org/RingAdjunction). A more known special case is a finite integral domain (http://planetmath.org/AFiniteIntegralDomainIsAField) — it is always a field and thus closed under the divisions.
|Title||finite ring has no proper overrings|
|Date of creation||2013-03-22 15:11:12|
|Last modified on||2013-03-22 15:11:12|
|Last modified by||pahio (2872)|