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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

\[ x = a_1 \ldots a_ m = b_1 \ldots b_ n \]

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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