# prime element is irreducible in integral domain

###### 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. ∎

Title prime element is irreducible in integral domain PrimeElementIsIrreducibleInIntegralDomain 2013-03-22 17:15:29 2013-03-22 17:15:29 me_and (17092) me_and (17092) 7 me_and (17092) Theorem msc 13G05 IrreducibleOfAUFDIsPrime