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单词 ClassicalRingOfQuotients
释义

classical ring of quotients


Let R be a ring. An element of R is called regularPlanetmathPlanetmathPlanetmath if it is not a right zero divisor or a left zero divisor in R.

A ring QR is a left classical ring of quotients for R(resp. right classical ring of quotients for R) if it satisifies:

  • every regular element of R is invertiblePlanetmathPlanetmath in Q

  • every element of Q can be written in the form x-1y (resp. yx-1) with x,yRand x regular.

If a ring R has a left or right classical ring of quotients, then it is unique up to isomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

If R is a commutativePlanetmathPlanetmathPlanetmath integral domainMathworldPlanetmath, then the left and right classical rings of quotients always exist – they are the field of fractionsMathworldPlanetmath of R.

For non-commutative rings, necessary and sufficient conditions are given by Ore’s Theorem.

Note that the goal here is to construct a ring which is not too different from R, but in which more elements are invertible. The first condition says which elements we want to be invertible. The second condition says that Q should just enough extra elements to make the regular elements invertible.

Such rings are called classical rings of quotients, because there are other rings of quotients. These all attempt to enlarge R somehow to make more elements invertible (or sometimes to make ideals invertible).

Finally, note that a ring of quotients is not the same as a quotient ringMathworldPlanetmath.

Titleclassical ring of quotients
Canonical nameClassicalRingOfQuotients
Date of creation2013-03-22 14:03:01
Last modified on2013-03-22 14:03:01
Ownermclase (549)
Last modified bymclase (549)
Numerical id5
Authormclase (549)
Entry typeDefinition
Classificationmsc 16U20
Classificationmsc 16S90
Synonymleft classical ring of quotients
Synonymright classical ring of quotients
Related topicOreCondition
Related topicExtensionByLocalization
Related topicFiniteRingHasNoProperOverrings
Definesregular
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