Definition 10.120.4. A *unique factorization domain*, abbreviated *UFD*, is a domain $R$ such that if $x \in R$ is a nonzero, nonunit, then $x$ has a factorization into irreducibles, and if

are factorizations into irreducibles then $n = m$ and there exists a permutation $\sigma : \{ 1, \ldots , n\} \to \{ 1, \ldots , n\} $ such that $a_ i$ and $b_{\sigma (i)}$ are associates.

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