I am studying ring theory and trying to solve the following problem:
Let $R$ be a commutative ring and $S$ be a multiplicative subset of $R$ not containing $0$. Let $P$ be a maximal element in the set of ideals of $R$ whose intersection with $S$ is empty. Prove that $P$ is a prime ideal.
I want to check this: is $S=R-P$ from the construction?
There is a general bijection $$ \{\text{ideals of $R$ disjoint with $S$}\}\leftrightarrow \{\text{ideals of $S^{-1}R$}\} $$ that preserves the inclusions and primality of ideals.
By maximality of $P$ among ideals disjoint from $S$ the ideal $S^{-1}P$ is maximal in $S^{-1}R$ hence prime. But then $P$ was prime by the correspondence.