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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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    It depends on whether you require that the identity in a subring agrees with the identity in the overring.2012-01-09
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    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!2012-01-09
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    "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.2012-01-09
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    Wow :) ${}{}{}{}$2012-01-09
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    @DylanMoreland The textbook from which the problem is drawn never requires a subring to have the same identity element; in fact, there are several examples given of subrings which have different identity elements. Thanks!2012-01-09
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    @Kurtis What textbook is that?2012-01-09
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    @BillDubuque Gallian's Contemporary Abstract Algebra2012-01-09
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    @Dylan: would not $\mathbb{Z} \otimes \mathbb{Z}_3$ do the trick in case of rings with identity?2012-01-10
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    @MarcOlschok That ring is $\mathbf Z/3$, isn't it?2012-01-10
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    @Dylan: yes, it is $\mathbb{Z}/(3)$. But I was writing nonsense and you were correct, there is no solution if rings with $1$ are assumed.2012-01-13
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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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