prime element is irreducible in integral domain

Theorem.

Every prime element of an integral domain is irreducible.

Proof.

Let $D$ be an integral domain, and let $a\in D$ be a prime element. Assume $a=bc$ for some $b,c\in D$.

Clearly $a\mid bc$, so since $a$ is prime, $a\mid b$ or $a\mid c$. Without loss of generality, assume $a\mid b$, and say $at=b$ for some $t\in D$.

If $1$ is the unity of $D$, then

$$1b=b=at=(bc)t=b(ct).$$

Since $D$ is an integral domain, $b$ can be cancelled, giving $1=ct$, so $c$ is a unit.∎