Proposition 17

Let A be a ring and $\mathfrak{p}_1, \ldots, \mathfrak{p}_n$ prime ideals of A. We write

(a) The ring $S^{-1}A$ is semi-local; if $q_1, \ldots, q_r$ are the distinct maximal elements (with respect to inclusion) of the set of $\mathfrak{p}_i$, the maximal ideals of $S^{-1}A$ are the $S^{-1}q_j$ ($1 \leq j \leq r$) and these ideals are distinct.

(b) The ring $A_{\mathfrak{p}_i}$ is canonically isomorphic to $(S^{-1}A)_{S^{-1}\mathfrak{p}_i}$ for $1 \leq i \leq n$.

(c) If A is an integral domain, then $S^{-1}A = \bigcap_1^n A_{\mathfrak{p}_i}$, in the field of fractions of A.

(a) The ideals of A not meeting S are the ideals contained in the union of the $\mathfrak{p}_i$ and hence in at least one of the $\mathfrak{p}_i$ (§ 1, no. 1, Proposition 2); the $q_j$ are therefore the maximal elements of the set of ideals not meeting S; consequently, the $S^{-1}q_j$ are the maximal ideals of $S^{-1}A$ by § 2, no. 5, Proposition 11 (ii).

(b) is a special case of § 2, no. 5, Proposition 11 (iii).

(c) Suppose that A is an integral domain. If $\mathfrak{p}_i \subset \mathfrak{p}_k$, then $A_{\mathfrak{p}_i} \supset A_{\mathfrak{p}_k}$; to prove (c), we may therefore suppose that no two $\mathfrak{p}_i$ are comparable. Then it follows from (a) and no. 3, Corollary 4 to Theorem 1 that $S^{-1}A = \bigcap_1^n (S^{-1}A)_{S^{-1}\mathfrak{p}_i}$; whence (c) by virtue of (b).

If A is an integral domain, so is $S^{-1}A$ and Proposition 17 then provides an example of a semi-local ring which is not a direct composition of local rings (cf. Chapter III, § 2, no. 13).