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单词 FiniteRingHasNoProperOverrings
释义

finite ring has no proper overrings


The regular elementsPlanetmathPlanetmath 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 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 (http://planetmath.org/RingAdjunction).  A more known special case is a finite integral domainMathworldPlanetmath (http://planetmath.org/AFiniteIntegralDomainIsAField) — it is always a field and thus closed under the divisions.

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更新时间:2025/5/4 15:01:39