Corollary 5

Let $A$ be a ring, $S$ a multiplicative subset of $A$ and $B$ a ring such that $i_A^S(A) \subset B \subset S^{-1}A$. Let $q$ be a prime ideal of $B$ such that the prime ideal $p = (i_A^S)^{-1}(q)$ of $A$ does not meet $S$ and let $p'$ be the prime ideal $S^{-1}p$ of $S^{-1}A$. Then $p' \cap B = q$.

Let $S' = i_A^S(S)$; a canonical isomorphism has been defined from ${S'}^{-1}B$ to $S^{-1}A$ (no. 1, Corollary 4 to Proposition 2); we identify these two rings by means of this isomorphism. As $q \cap S' = \varnothing$, $q' = {S'}^{-1}q$ is the unique prime ideal of $S^{-1}A = {S'}^{-1}B$ such that $q' \cap B = (i_A^S)^{-1}(q') = q$ (Proposition 11 (ii)), whence $(i_A^S)^{-1}(q') = p$; consequently $q' = p'$ (Proposition 11 (ii)).

In the notation of Corollary 5, there are canonical isomorphisms of $A$, and $B_q$ onto $(S^{-1}A)_q$. (Proposition 11 (iii)) and hence a canonical isomorphism $A, \to B_q$.