Proof of that in an integral domain, every prime element is irreducible. I would like to prove that in an integral domain $R$, every prime element $p$ is irreducible. I understand the case where $p = ab$ but the textbooks I have read do not address the case where $p \neq ab$, i.e.,$px = ab$.
I was wondering why they do not discuss the case where $px = ab$.

Proof:
suppose P is a prime element of an integral domain R, and p=ab
we know that, P|P => P|ab => P|a or P|b
              P|a and p=ab or a|P => a is unit 
similarly P|b and p=ab ir b|p => b is unit,
so either a is a unit or b is a unit, hence p must be irreducible element,
