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.

  • $\begingroup$ As what should $I$ and $2\mathbb Z$ be isomorphic? $\endgroup$ – Hagen von Eitzen Jun 5 '15 at 13:44
  • $\begingroup$ I dont quite understand both your questions. I checked yhe answer should be Z X Z has an ideal I = 2Z X {0} . Which is isomorphic to 2Z $\endgroup$ – italy Jun 5 '15 at 13:46
  • $\begingroup$ The question is justified. The ideal $\mathbb Z \times 0 \subseteq \mathbb Z \times \mathbb Z$ is also isomorphic to $2\mathbb Z$ as $\mathbb Z$-modules. $\endgroup$ – Thomas Poguntke Jun 5 '15 at 21:34

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.

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