# Ring equals max-localization intersection

From Commalg

## Statement

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

## Proof

Let . Let be the ideal comprising those for which . is essentially the ideal of all possible *denominators* of fractions for in terms of elements of .

The claim is that . Suppose not. Then is a proper ideal of ,and since every proper ideal is contained in a maximal ideal, we can find a maximal ideal containing . But since , can be written as where . Clearly , and this contradicts .