complete ring of quotients

Consider a commutativePlanetmathPlanetmathPlanetmath unitary ring R and set

𝒮:={HomR(I,R):I is dense in R}

(here HomR(I,R) is the set of R-module morphismsMathworldPlanetmathPlanetmath from I to R) and define A:=B𝒮B.

Now we shall assign a ring structureMathworldPlanetmath to A by defining its addition and multiplication. Given two dense ideals I1,I2R and two elements fiHomR(Ii,R) for i{1,2}, one can easily check that I1I2 and f2-1(I1) are nontrivial (i.e. they aren’t {0}) and in fact also dense ideals so we define

f1+f2HomR(I1I2,R) by (f1+f2)(x)=f1(x)+f2(x)

f1*f2HomR(f2-1(I1),R) by (f1*f2)(x)=f1(f2(x))

It is easy to check that A is in fact a commutative ring with unity. The elements of A are called .

There is also an equivalence relationMathworldPlanetmath that one can define on A. Given fiHomR(Ii,R) for i{1,2}, we write


(i.e. f1 and f2 belong to the same equivalence classMathworldPlanetmath iff they agree on the intersectionMathworldPlanetmathPlanetmath of the dense ideal where they are defined).

The factor ring Q(R):=A/ is then called the complete ring of quotients.


RT(R)Q(R), where T(R) is the total quotient ring. One can also in general define complete ring of quotients on noncommutative rings.


  • Huckaba J.A. Huckaba, ”Commutative rings with zero divisorsMathworldPlanetmath”, Marcel Dekker 1988
Title complete ring of quotients
Canonical name CompleteRingOfQuotients
Date of creation 2013-03-22 16:20:29
Last modified on 2013-03-22 16:20:29
Owner jocaps (12118)
Last modified by jocaps (12118)
Numerical id 17
Author jocaps (12118)
Entry type Definition
Classification msc 13B30
Related topic CompleteRingOfQuotientsOfReducedCommutativeRings
Related topic EpimorphicHull
Defines fraction of rings
Defines complete ring of quotients