Show by example that not every ideal of a subring $S$ of a ring $R$ need be of the form $I\cap S$ for some ideal $I$ of $R$
If $I$ is an ideal of $R$ and $S$ is a subring of $R$, then it can be easily shown that $I\cap S$ is an ideal of $S$. However, what is an example when this is not necessary.