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> Let $R$ be a Euclidean domain. Then any subring $S$ of $R$ is a unique factorization domain. Is this statement true or not?
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my idea: $R$ is a Euclidean domain implies that $R$ is a principal ideal domain. Hence $R$ is a unique factorization domain. Hence for any $r\in R$, there is a unique expression $r=u\cdot\Pi\_{i=1}^n p\_i$, where $u$ is a unit in $R$ and $p\_i$ is irreducible in $R$. The expression is unique up to multiples of units in $R$ to each of the factor. This implies that for any $s\in S$, there is a unique expression $s=u\cdot\Pi\_{i=1}^n p\_i$, where $u$ is a unit in $R$ and $p\_i$ irreducible in $S$. The expression is unique up to multiples of units in $R$ to each factor. But a unit in $R$ is not neccesarily a unit in $S$. What can I do?