Subring of $\Bbb Z_{18}$ with unity

Question

Need help finding subrings $A$ and $B$ of $\Bbb Z_{18}$ in which $A$ and $B$ are rings with unity, $B$ is a subring of $A$, but the unity of $B$ is not the same as the unity of $A$.

Answer 1

Hint. Consider $\{0,9\}$. Show that under the operations of addition modulo $18$ and multiplication modulo $18$, this is a ring with unity.

Answer 2

Hints:

If the ring you're looking for has identity $e$, then $e$ is idempotent: that is, $e^2=e$. In any ring $R$ with identity and an idempotent $e\neq 0$, $eRe$ is a subset that's a ring which has identity $e$.

So the question is now "What are the idempotents of $\Bbb Z_{18}$?