Euclid's Lemma for Irreducible Elements

Lemma
Let $\struct {D, +, \times}$ be a Euclidean domain whose unity is $1$.

Let $p$ be an irreducible element of $D$.

Let $a, b \in D$ such that:
 * $p \divides a \times b$

where $\divides$ means is a divisor of.

Then $p \divides a$ or $p \divides b$.

Proof
Let $p \divides a \times b$.

Suppose $p \nmid a$.

Then from the definition of irreducible:
 * $p \perp a$

Thus from Euclid's Lemma for Euclidean Domains:
 * $p \divides b$

Similarly, if $p \nmid b$:
 * $p \divides a$

So:
 * $p \divides a b \implies p \divides a$

or:
 * $p \divides b$

as we needed to show.

Also see

 * Euclid's Lemma for Prime Divisors, for the usual statement of this result, which is this lemma as applied specifically to the integers.