It will be shown now that every irreducible element in a factorial ring is also prime.
Let R be a factorial ring.
then let $a\in R$ be irreducible and $x,y \in R$ so that $a| xy$. It is now to show that $a|x$ or $a|y$. It is assumed that x and y are not units. Let $x=x_{1}…x_{r}, y=y_{1}…y_{s}$ be partitions into irreducible elements. Then it follows that $a|(x_{1}…x_{r}y_{1}…y_{s})$. Since this is unique it means that a as a irreducible element can be associated with a $x_{i}$ or a $y_{i}$. That's why $a|x$ or $a|y$ and a is a prime.
Tell me if this proof is correct. Please
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