Simple question about ideals and quotient rings

Question

This one should be easy, so easy in fact that it should be possible to get the answer by plain googling, if only I could figure out the right terms to google.

Let's say that you have ideals $I_1 \subset I_2$ of a ring $R$. Is it then true that $R/I_2$ is isomorphic to a subring of $R/I_1$?

If so, where can I find a proof?

If not, what counterexample is there to offer?

Thanks in advance!

Answer 1

The ring $R/I_2$ is a quotient of $R/I_1$, but is not a subring in general. Take for instance $R = \mathbb{Z}$, $I_1 = 0$ and $I_2 = 2\Bbb{Z}$. Then $\Bbb{Z}/2\Bbb{Z}$ is not isomorphic to any subring of $\Bbb{Z}$.