MathGloss

An integral domain $R$ is a unique factorization domain if

1. Every nonzero, non-unit $r\in R$ can be written as a product of irreducibles; and
2. Whenever $p_1\cdots p_n=q_1\cdots q_m$ for $p_i,q_j$ irreducible and $m,n\in\mathbb N$, then $m=n$ and there is a relabeling/reordering of the $q_j$ such that $q_i$ is associate to $p_i$ for all $1\leq i\leq n$.

In particular, for every association class of irreducibles, we can choose a representative, and let $P$ be the collection of such representatives so that every $r\in R$ can be written uniquely as $$r= u\prod_{p\in P} p^{e_p}$$ for $u$ a unit and all but finitely many of the $e_p$ equal to zero.

Wikidata ID: Q1052579

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