Let $R$ be a commutative ring with unity. If a subring $S$ of $R$ is an integral domain containing the unity of $R$ (that isn’t $\{0,1\}$), does this imply that $R$ is too an integral domain? I tried to find a counterexample but I did not find one:
I figured maybe the subring of $Z_{10}$, $(2)$, would work since I figured it would be isomorphic to $Z_5$ (and thus an integral domain), but then I realized it has no multiplicative identity so that’s a no.
Any help would be appreciated.
Best Answer
Another good class of rings for examples is polynomial rings and their quotients. For this problem, consider the subring $\mathbb C$ inside $\mathbb C[x]/\langle x^2\rangle$.