Localization, The Intersection of Fraction Rings

The Intersection of Fraction Rings

Let R be an integral domain. Thus R embeds in each S inverse of R, embeds in the fraction field F. The intersection of all these intermediate fraction rings produces R. Specifically, the intersection of the localizations R_P yields R. Beyond this, the intersection of the localizations over maximal ideals produces R.

Let F be the fraction field of R. Let x, a member of F, lie in the intersection of the localizations of R.

Let J be the ideal of R that maps x into R. In other words, y ∈ J iff yx ∈ R.

If you write x as a fraction, then the denominator lies in J.

If J is all of R then J contains 1, and x is already in R, so assume J is a proper ideal of R. Thus J is contained in a maximal ideal M in R.

If x can be represented by the fraction a/b, then b lies in J, which lies in M.

Localize about M, and suppose x is represented by some a/b in R_M. We just showed b lies in M, hence a/b is not part of R_M. This is a contradiction, hence x lies in R after all, and the intersection of localizations resurrects R.