$\lhd$ will stand for "is an ideal of" in this post.
Let $R$ be a commutative ring, $J\lhd I\lhd R$. Does it follow that $J\lhd R?$
I don't think it does, but I'm having difficulty finding a counterexample. An example for a non-commutative $R$ is here.
I've tried several things, but at random really, so I don't think it makes sense to post it here.