The Stacks project

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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