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# finite ring has no proper overrings

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.

## Mathematics Subject Classification

13G05*no label found*

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