[[Integral domain]]
# All primes are irreducible in an integral domain

Let $D$ be an [[integral domain]] and $\pi \in D$ be a [[prime element]].
Then $\pi$ is also an [[irreducible element]]. #m/thm/ring 

> [!check]- Proof
> Suppose $\pi = ab$ with $a,b \in R$.
> Then $\pi \mid ab$, so without loss of generality $a = \pi u$.
> Thus $\pi 1 = ab = \pi u b$ so by the cancellation property $ub = 1$,
> whence $b$ is a unit.
> Therefore $\pi$ is irreducible. <span class="QED"/>

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