question involved ideal and algebra

Question

Every ring here is commutative with unit. $R$ is ring, let $S$ be an $R$-algebra. $J\subset S$ is an ideal, and $S/J\cong R$. Prove for any ideal $I\subset R$, we have $IJ=IS\cap J$.

It's easy to see $IJ\subset IS\cap J$, but how to prove the other direction?

Answer 1

This is true, more generally, if $S/J$ is flat over $R$ (instead of "isomorphic to $R$"). See Bourbaki, Algèbre Commutative, Chap.I, §2, n.6, corollaire (i) de la proposition 7.