Let $R$ be a commutative Noetherian ring with unit and $S$ a flat $R$-algebra. Does the [going-up theorem](http://en.wikipedia.org/wiki/Going_up_and_going_down#Going-up) hold between $R$ and $S$?



Does the going-up theorem hold between flat algebras?
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.


