In a commutative ring $A$ with $1$ having a subset $S$ and some element $x \notin S$, does there always exist a prime ideal containing $S$ but not $x$? I am trying to solve the $\Rightarrow$ part of (iii). If this is not true always,can you help me figure out a solution? Thanks in advance.
(Edit) I could not understand the solution given in the link.
