We know that an integral domain is a commutative ring with unity and no zero-divisors. It is obvious that if $R$ is an integral domain and $S$ is a subring of $R$ that $S$ must also be commutative, and if $a,b\in S$ and $ab=0$, then $a=0$ or $b=0$. But I'm having trouble proving that $S$ has unity, does the unity of $R$ have to be the unity of $S$?
Also what's a good counterexample for a subring of a field that is not a field.