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
.