Is this obvious? I cannot see that this is true. The converse is fairly obvious though. I tried to show $(x)$ is a maximal ideal and try the quotient but failed. I will appreciate any help.
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3Duplicate of http://math.stackexchange.com/questions/1297981/show-that-if-rx-is-euclidean-domain-then-r-is-a-field. – lhf Jun 13 '16 at 00:35
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@Anton, the OP meant a general ring $R$, not the reals $\mathbb R$. – lhf Jun 13 '16 at 01:33
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@Ihf Ohh, thanks, that was my inattention, I'm sorry. – Anton Grudkin Jun 13 '16 at 01:40
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More generally, $R[x]$ is a PID iff $R$ if a field.
Hint: Take $r \in R \setminus 0$ and consider the ideal $(r,x)$.
lhf
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Oh i see the ideal is the whole ring...... Thank you for your neat answer. – Mathcho Jun 13 '16 at 01:04