The regular elements 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 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 $T$ contains 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. A more known special case is a finite integral domain -- it is always a field and thus closed under the divisions.