Let $R$ be a ring and $I$ is it true that:"If $R$ is not an integral domain then also $R/I$ isn't an integral domain"?``

Question

The question: Let $R$ be a ring and $I$ be an ideal of $R$. Is it true that if $R$ is not an integral domain then also $R/I$ isn't an integral domain?

My attempt at solution is the following:

I don't think the claim is true. As a counterexample I take $R=6\mathbb{Z}$ and $I=2\mathbb{Z}$ I believe that $R/I=3\mathbb{Z}$ which is an integral domain, while $R$ isn't an integral domain since it has zero divisors.

I am not sure, if I got it right.

Any help?

Thanks!

Answer 1

Choose, $R=\Bbb{Z_6}$ and $I =\{0, 3\}$

Now, $R/ I \cong Z_3$ which is an integral domain.

Hence, $R/ I$ integral domain doesn't imply $R$ is an integral domain.

Answer 2

In any non trivial ring $R$ there exists a maximal ideal $M$. Then $R/M$ is a field, so in particular it is an integral domain.