Ring equals max-localization intersection

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Statement

Let R be a commutative unital ring and K(R) its total quotient ring. For each maximal ideal M of R, let R_M denote the localization of R at M viewed as a subring of K(R). Then:

R = \bigcap R_M

Proof

Let a \in \bigcap R_M. Let I be the ideal comprising those x \in R for which ax \in R. I is essentially the ideal of all possible denominators of fractions for a in terms of elements of R.

The claim is that I = R. Suppose not. Then I is a proper ideal of R,and since every proper ideal is contained in a maximal ideal, we can find a maximal ideal M containing I. But since a \in R_M, a can be written as p/q where q \notin M. Clearly q \in I, and this contradicts I \le M.