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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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    $\begingroup$ It depends on whether you require that the identity in a subring agrees with the identity in the overring. $\endgroup$ – Qiaochu Yuan Jan 9 '12 at 2:44
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    $\begingroup$ Note that for some authors, "subring" is supposed to imply "has the same identity element", which makes this exercise impossible. If you don't have that restriction, then this looks great! $\endgroup$ – Dylan Moreland Jan 9 '12 at 2:44
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    $\begingroup$ "Subring" should probably be interpreted in the sense which does not require the small ring and the big one to share the identity element. It never hurts to be explicit about that. $\endgroup$ – Mariano Suárez-Álvarez Jan 9 '12 at 2:45
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    $\begingroup$ Wow :) ${}{}{}{}$ $\endgroup$ – Mariano Suárez-Álvarez Jan 9 '12 at 2:45
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    $\begingroup$ @MarcOlschok That ring is $\mathbf Z/3$, isn't it? $\endgroup$ – Dylan Moreland Jan 10 '12 at 23:08
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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 $3$.

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