Lemma 33.37.12. Let A be a ring. Let \mathfrak p_1, \ldots , \mathfrak p_ r be a finite set of a primes of A. Let S = A \setminus \bigcup \mathfrak p_ i. Then S is a multiplicative system and S^{-1}A is a semi-local ring whose maximal ideals correspond to the maximal elements of the set \{ \mathfrak p_ i\} .
Proof. If a, b \in A and a, b \in S, then a, b \not\in \mathfrak p_ i hence ab \not\in \mathfrak p_ i, hence ab \in S. Also 1 \in S. Thus S is a multiplicative subset of A. By the description of \mathop{\mathrm{Spec}}(S^{-1}A) in Algebra, Lemma 10.17.5 and by Algebra, Lemma 10.15.2 we see that the primes of S^{-1}A correspond to the primes of A contained in one of the \mathfrak p_ i. Hence the maximal ideals of S^{-1}A correspond one-to-one with the maximal (w.r.t. inclusion) elements of the set \{ \mathfrak p_1, \ldots , \mathfrak p_ r\} . \square
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