finite ring has no proper overrings

The regular elementsPlanetmathPlanetmath of a finite commutative ring R are the units of the ring (see the parent ( of this entry).  Generally, the largest overring of R, the total ring of fractionsMathworldPlanetmath T, is obtained by forming S-1R, the extension by localization, using as the multiplicative set S the set of all regular elements, which in this case is the unit group of R.  The ring R may be considered as a subring of T, which consists formally of the fractions  as=as-1  with  aR  and  sS.  Since every s has its own group inverse s-1 in S and so in R, it’s evident that T no other elements than the elements of R.  Consequently,  T=R,  and therefore also any overring of R coincides with R.

Accordingly, one can not extend a finite commutative ring by using a localizationMathworldPlanetmath.  Possible extensionsPlanetmathPlanetmathPlanetmath must be made via some kind of adjunction (  A more known special case is a finite integral domainMathworldPlanetmath ( — it is always a field and thus closed under the divisions.

Title finite ring has no proper overrings
Canonical name FiniteRingHasNoProperOverrings
Date of creation 2013-03-22 15:11:12
Last modified on 2013-03-22 15:11:12
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Result
Classification msc 13G05
Related topic ExtensionByLocalization
Related topic ClassicalRingOfQuotients
Related topic AFiniteIntegralDomainIsAField
Related topic RingAdjunction
Related topic FormalPowerSeries