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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.
Usually one requires a subring of a unital ring to contain the unit. If you remove this requirement, the result does not hold. For example, $\mathbb{Z}$ is an integral domain, but if we do not require subrings to contain the unit, $2\mathbb{Z}$ is a subring which is not an integral domain.
As for your second question, $\mathbb{Z}$ is a subring of the field $\mathbb{Q}$, but is not itself a field.
