Irreducible implies prime (PID)

Statement

Verbal statement

In a principal ideal domain, any irreducible element is a prime element.

Symbolic statement

Let $R$ be a principal ideal domain and $p$ an irreducible element in $R$. Then, if $p|ab$, then $p|a$ or $p|b$.

Proof

Suppose $p|ab$ and $p \not | a$. We need to prove that $p|b$.

Then, the ideal generated by $p$ and $a$ is principal, and is generated by a factor of both $p$ and $a$. Since $p$ is irreducible, the only possibility for this is the whole ring. Thus:

$1 = ax + py$

for suitable choices of $x,y \in R$.

Multiplying both sides by $b$, we get:

$b = abx + pby$

Since $p|ab$, $p$ divides the entire right-hand-side, and hence $p|b$.