Not every ideal of a subring $S$ of $R$ is of the form $I \cap S$ for $I \lhd R$

Question

I just got done showing that if $I \lhd R$ and $S \leq R$ then $I \cap S \lhd S$. I'm now looking for an example to show that not every ideal of a subring $S$ of $R$ is of the form $I \cap S$ for $I \lhd R$.

I tried finding examples in matrix rings to no avail.. If somebody could offer me a new perspective it would be much appreciated!

Answer 1

Let $S=\mathbb{Z}$ and $R=\mathbb{R}$. The only ideals of $R$ are $(0)$ and the whole ring. The ideal $(2)$ of $S$ is not the intersection of either of those ideals with $S$.