If $\phi :R_1\to R_2$ is an isomorphism of commutative rings and $I_1$ is an ideal of $R_1$ then $\phi (I_1) = I_2$ is an ideal of $R_2$ and furthermore $R_1/I_1\simeq R_2/I_2$. Define $\phi: 2\mathbb{Z}\to \mathbb{Z}$ by $\phi(x) = \tfrac{x}{2}$. This is an isomorphism. Let $I_1 = 8\mathbb{Z}$, this is an ideal of $2\mathbb{Z}$. Note that $I_2 = \phi (I_1) = 4\mathbb{Z}$. Thus, $2\mathbb{Z}/I_1 \simeq \mathbb{Z}/I_2 = \mathbb{Z}_4$.