Remark

The intersection C of the rings $A_i'$ of the valuations $v_i'$ ($i \in I$) is the integral closure of $A$ in L (§ 1, no. 3, Corollary 3 to Theorem 3); it follows moreover from (c) and § 7, no. 1, Propositions 1 and 2 that C is a semi-local ring, that its maximal ideals are the intersections $m_i = C \cap m(A_i')$ and that $A_i' = C$, for all $i \in I$.