Does the going-up theorem hold between flat algebras?

Question

Let $R$ be a commutative Noetherian ring with unit and $S$ a flat $R$-algebra. Does the going-up theorem hold between $R$ and $S$?

Answer 1

For a counterexample, take $$R={\mathbb Z}\qquad S=R[x]\qquad P=(1+2x)\subset S$$.
Then $P\cap R=(0)\subset (2)$, so if going-up holds, then there is a prime $Q$ in $S$ containing $(1+2x)$ and such that $Q\cap R=(2)$. But then $Q$ contains $2$, so $Q$ contains $(2,1+2x)=S$, contradiction.