Corollary 2

Let $A$ be a ring and $S$ a multiplicative subset of $A$. For every ideal $a$ of $A$ not meeting $S$ there exists a prime ideal containing $a$ and not meeting $S$.

$S^{-1}a \neq S^{-1}A$ (Remark) and hence there exists a maximal ideal of $S^{-1}A$ containing $S^{-1}a$ (Algebra, Chapter I, § 8, no. 7, Theorem 2) and the corollary follows from Proposition 11 (ii).