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Any irreducible element of a factorial ring $D$ is a prime element of $D$.

Proof. Let $p$ be an arbitrary irreducible element of $ D$. Thus $ p$ is a non-unit. If $ ab \in (p)\smallsetminus\{0\}$, then $ ab = cp$ with $ c \in D$. We write $ a,\,b,\,c$ as products of irreducibles: $\displaystyle a \;=\; p_1\cdots p_l, \quad b \;=\; q_1\cdots q_m, \quad c \;=\; r_1\cdots r_n$. Here, one of those first two products may me empty, i.e. it may be a unit. We have $$\displaystyle p_1\cdots p_l\,q_1\cdots q_m \;=\; r_1\cdots r_n\,p\tag{1}$$

Due to the uniqueness of prime factorization, every factor $ r_k$ is an associate of certain of the $l+m$ irreducibles on the left hand side of $(1)$. Accordingly, $p$ has to be an associate of one of the $ p_i$'s or $ q_j$'s. It means that either $ a \in (p)$ or $ b \in (p)$. Thus, $ (p)$ is a prime ideal of $ D$, and its generator must be a prime element.

It may be too simple, but why $ a \in (p)$ instead of $p_1 \in (p)$? Is it because $p$ has to be an associate of one of the $ p_i$'s or $ q_j$'s? Let's say $p_2$ is an associate of $p$. So, $p_2=pw$, $w\in R$. Since $a=p_1p_2\cdots p_l$ then $a=p_1pwp_3\cdots p_l$ and $a=p(p_1p_3\cdots p_lw)$, $p_1p_3\cdots p_lw \in R$ so $a$ is divisible by $p$ hence $a\in (p)$ ?

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Your reasoning looks correct to me. – Zach L. Dec 13 '12 at 15:12
Yes, more simply $\rm\: p\mid p_i\mid a,\:$ i.e. it follows by transitivity of "divides" (or "contains", if expressed using ideals) – Bill Dubuque Dec 13 '12 at 15:32

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