I have to prove the following that every associate of an irreducible element is irreducible.
I am working in an integral domain so I will use the theorem which states that whenever $p$ is a non zero non unit element and when $p=rs$ then either $r$ or $s$ is a unit.
Let $p$ be an irreducible element. with $p=rs$ then $r \ or \ s \ is \ a \ unit$
Now let the associate of the irreducible element $p$ be $a$ s.t. $a \in R$ then $a=pu$ where u is a unit. Hence $a=rsu$ with either r or s being units.
If $r$ is a nonunit then $a=s$ and then $s$ must be a unit from theorem.
If $s$ is a nonunit then $a=r$ where r must be a unit from theorem.
So from the theorem $a$ the associate must be irreducible?