Quotient with divisors of zero

Question

Is there a ring $R$ with divisors of zero which have an ideal $I$ (non-null neither equal to $R$) that, the quotient $R/I$ also has divisors of zero?

Answer 1

Think about the residue class ring $R={\Bbb Z}_8$. The multiples of 4 form an ideal in $R$. Then $R/\langle 4\rangle$ is isomorphic to ${\Bbb Z}_4$ which also has zero divisors.

Answer 2

Anytime you take a proper ideal $I$ of $R$ (a commutative ring) that is not prime, the quotient ring $R/I$ has nontrivial zero divisors.

Now take your favorite ring, for instance $\mathbb{Z}$; take two ideals $I$ and $J$ that are not prime and with $I\subsetneq J\subsetneq\mathbb{Z}$. Set $R=\mathbb{Z}/I$, so $R$ has zero divisors; take the ideal $J'=J/I$ in $R$ and note that $R/J'\cong R/J$ has zero divisors because $J$ is not prime.

Answer 3

Let be $R=\mathbb Z_{12}$ and $I=\{\overline{0},\overline{6}\}$, then $R/I= \{\overline{0}+I,\overline{1}+I,\overline{2}+I,\overline{3}+I,\overline{4}+I,\overline{5}+I \} \cong \mathbb Z_6$ which has zero divisors, $(\overline{2}+I)(\overline{3}+I)=I$.