> [!NOTE] Lemma (Prime Elements of Integral Domain are Irreducible) > Let $R$ be an [[Integral Domain|integral domain]]. If $p\in R$ be [[Prime Elements of Integral Domain|prime]] then $p$ is [[Irreducible Elements of Integral Domain|irreducible]]. ###### Proof that Prime Elements of Integral Domain are Irreducible Suppose $p=ab$. Then $p\mid ab$ so $p\mid a$ or $p\mid b$ since $p$ is prime. WLOG suppose $p\mid a$, that is $a=pv$ for some $v\in R$. Then $p=pvb$. Applying the [[Cancellation Law for Integral Domains|cancellation law]] yields $1=vb$ and so $v$ is a unit. Hence $p$ is irreducible. $\square$