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What would be the ideals of the real numbers?

I have to figure out the ideals of a ring R, which is a product of integers and reals like so:

$R = \mathbb{Z}\times \mathbb{R} \times \mathbb{Z}$

Are the ideals simply 0 and $\mathbb{R}$?

Dazzler95
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  • I am going to close this as a duplicate of http://math.stackexchange.com/q/101157/29335 unless you want to change the question to ask about the ring $\mathbb Z\times \mathbb R\times \mathbb Z$. To answer your last question: yes. – rschwieb Nov 23 '16 at 21:04
  • @rschwieb At a second thought I think your link is better for closing this as a duplicate. – user26857 Nov 23 '16 at 21:15
  • @user26857 All bases are covered as it is, probably don't need to overoptimize now. – rschwieb Nov 23 '16 at 21:24
  • The explanation for how a ring is a field iff its only ideals are 0 and itself is very helpful, so thanks. – Dazzler95 Nov 23 '16 at 21:26

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Hint: If $R$ and $S$ are rings with unity, then the ideals of $R\times S$ are of the form $I\times J$ where $I$ is an ideal of $R$ and $J$ is an ideal of $S$. Since the product of two rings with unity is a ring with unity, ideals of $R\times S\times T$ with $R$, $S$, $T$ all unital rings will be of the form $I\times J\times K$ where $I$, $J$, and $K$ are ideals of $R$, $S$, and $T$, respectively. $\Bbb R$ is a field, so the only ideals are $(0)$ and $\Bbb R$. Then it remains to classify ideals of $\Bbb Z$.

Stahl
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