Let $R$ be a Noetherian commutative ring with unity, and $S=R[x_1,\dots,x_r]$. Let $f\in S$ and suppose that the ideal generated by the coefficients of $f$ is $R$. How to show that $f$ is a nonzerodivisor on $S/IS$ for every ideal $I$ of $R$?
What I tried:
By Noetherianity, I can pick an ideal $I$ maximal among those ideals $J$ such that $f$ is a zerodivisor on $S/JS$. If I can show that $I$ is prime, then I am done. However, I did not get anything out along this way.
EDIT: Actually, this is Exercise 6.4 in Eisenbud. I can prove the forward direction, and the backward direction needs the above proposition.