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I have seen that if you have a ring $R$ which is also a field $F$ then its corresponding polynomial ring $R[x]$ is an Euclidean domain.

So I am looking for counter examples of the converse that when $R[x]$ is an Euclidean domain then R need not be a field. I am not really sure how to look for these examples. Maybe there is some trivial ones that I have overlooked? or some non trivial examples ?

GNUSupporter 8964民主女神 地下教會
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    It is in fact true that $R[x]$ is a principal ideal domain if and only if $R$ is a field (and thus if and only if $R[x]$ is an Euclidean domain). See proofwiki. It has already been asked here as well, but I'm still loking for an appropriate duplicate. –  Apr 24 '18 at 09:57
  • thanks I see now –  Apr 24 '18 at 10:13

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