[[Ring theory MOC]]
# Unique factorization domain

A **unique factorization domain** or **UFD** $R$ is an [[integral domain]] such that every nonzero element $x \in R$ has a **factorization** as a product of [[irreducible element|irreducible elements]],
unique up to units and the order of factors. #m/def/num 
$$
\begin{align*}
x = q_{1} \cdots q_{r}
\end{align*}
$$
Every UFD is also a [[GCD domain]].

## Equivalent characterizations

Let $R$ be an [[integral domain]].
The following are equivalent:

1. $R$ is a UFD;
2. Every [[irreducible element]] in $R$ is [[Prime element|prime]] and $R$ satisfies the [[Noetherian ring#^N2]] on [[Principal ideal|principal ideals]].[^2009] ^U2

  [^2009]: 2009\. [[Sources/@aluffiAlgebraChapter02009|Algebra: Chapter 0]], § V.2.2, p. 253

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