a prime element of the integral domain is irreducible

Let $a$ be a prime element of the integral domain R.Then $a$ is irreducible.
Proof:$a$ is a prime element of the intergral domain R,which means that $a$ is not a unit,nor it is 0,and $\forall p,q\in R$,if $ac=pq$,then $\exists c_1\in R$ such that $ac_1=p$ or $ac_1=q$.Now let $c$ be 1.If $a$ is reducible,which means that $a=mn$(m,n are not units),then $a1=mn$,so $ac_1=m$ or $ac_1=n$.For example,if $ac_1=m$,then $ac_1n=a$,so $c_1n=1$,which means that $n$ is a unit.This is a contradiction.