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This is a homework question that I'm either not thinking through all the way, or I'm overcomplicating the issue. It reads

Give an example of a ring that contains a subring isomorphic to $\mathbb{Z}$ and a subring isomorphic to $\mathbb{Z}_3$.

My quick answer is that $\mathbb{Z}_3 \oplus \mathbb{Z}$ is such a ring. We can take $R = \{(a,0) | a \in \mathbb{Z}_3\}$ to be a subring isomorphic to $\mathbb{Z}_3$ and $S = \{(0,a) | a \in \mathbb{Z}\}$ to be a subring isomorphic to $\mathbb{Z}$.

Is there something crucial I'm missing here, or is the problem really that simple?

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    @QiaochuYuan Another answer of yours in the comments. Again a link to the [chat](http://chat.stackexchange.com/rooms/9141/the-crusade-of-answers).2013-06-09

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There's nothing crucial you're missing, although you should be aware that the convention that a subring need not share its identity with the overring is not standard. With that convention, the problem is impossible, since $\mathbb{Z}$ is a subring of a ring iff it has characteristic $0$ and $\mathbb{Z}/3\mathbb{Z}$ is a subring if a ring iff it has characteristic $3$.