Find two rings with unity $R$ that have an ideal $I$ isomorphic to $2\mathbb{Z}$. Identify the ring $R/I$ in each case.
I know $2\mathbb{Z}$ is an ideal of $\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z} \simeq \mathbb{Z}_2$.
So thats one, I guess? How to proceed from here?
Any help appreciated. Thanks.
Take a finite direct product $R=\mathbb{Z}\times \cdots \times \mathbb{Z}$ of length $n$. Then $I=0\times \cdots \times 2\mathbb{Z} \times \cdots \times 0$ is an ideal in $R$, with quotient isomorphic to $\mathbb{Z}\times \cdots \times \mathbb{Z}/2\mathbb{Z}$. There are as many possibilities for $R$ as you can place the factor $2\mathbb{Z}$, i.e., $n$ possibilities.