# finite ring has no proper overrings

The regular elements of a finite commutative ring $R$ are the units of the ring (see the parent (http://planetmath.org/NonZeroDivisorsOfFiniteRing) of this entry).  Generally, the largest overring of $R$, the total ring of fractions $T$, is obtained by forming $S^{-1}R$, 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  $\frac{a}{s}=as^{-1}$  with  $a\in R$  and  $s\in S$.  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 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 FiniteRingHasNoProperOverrings 2013-03-22 15:11:12 2013-03-22 15:11:12 pahio (2872) pahio (2872) 10 pahio (2872) Result msc 13G05 ExtensionByLocalization ClassicalRingOfQuotients AFiniteIntegralDomainIsAField RingAdjunction FormalPowerSeries