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?
2026-03-31 14:30:47.1774967447
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?
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Hint: Fields are EDs and non-UFDs can have residue fields.