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Show that a non-empty subset $S ⊂ R$ is a subring of $R$ if for all $r, s ∈ S$ we have $r − s ∈ S$ and $rs ∈ S$. (This makes it easier to verify a set is a ring, if you know the set lives in a larger ring.)

Also show that $\mathbb Z[√2] ∪ \mathbb Z[i]$ is not a subring of $\mathbb C$.

I'm clueless. How do I start this?

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  • $\begingroup$ Your first question about s completely answered by previous posts like this one math.stackexchange.com/q/40677/29335 if you'd only search for them. Then you could skip that and get straight to the second question you squeezed in. $\endgroup$ – rschwieb Feb 26 '16 at 4:31
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Hints: For the first question: if $r-s \in S$ for any $r,s \in S$ then in particular $-s = 0-s \in S$ for any $s \in S$ and $r + s = r - (-s) \in S$ for all $r,s \in S$.

For the second question: is $\sqrt 2 + i$ in $\mathbb{Z}[\sqrt 2] \cup \mathbb{Z}[i]$?

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