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  $\frac{a}{s}=a{s}^{-1}$  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
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