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Let $R$ be an integral domain and $I$ be a prime ideal of $R$. If $R/I$ is a Euclidean domain, will $R$ be a unique factorization domain?

I have no idea to prove or disprove this... should I prove or disprove?

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My immediate reaction to questions worded like this is to say that it depends on $R$ and $I$. I can think of more than one way to interpret the question, and you might find it helpful to phrase it more formally. –  Derek Holt Dec 28 '13 at 9:14
    
To be more specific, are you assuming that $R/I$ is a Euclidean domain for all prime ideals $I$ of $R$, or that there exists an ideal $I$ of $R$ such that $R/I$ is a Euclidean Domain? –  Derek Holt Dec 28 '13 at 14:03
    
Do you know any nonunique factorization domains? –  Bill Dubuque Dec 28 '13 at 17:36

1 Answer 1

Hint: Fields are EDs and non-UFDs can have residue fields.

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should I prove or disprove? –  RSQ Dec 28 '13 at 8:01
    
Understand my hint and you will know whether to prove or disprove (and how). –  anon Dec 28 '13 at 8:03
    
@RenShiquan Here's a (hopefully not overkill) complementary hint to this one. Sometimes it's good to consider extremes. Let $R$ be as bad as possible, but make sure that $R/I$ is as good as possible. This is the direction of anon's hint –  rschwieb Dec 28 '13 at 12:07

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