Let $R$ be a commutative ring, and let $a \in R$ be an irreducible element. Prove that $a$ is not a zero divisor.
I need help proving this. I know that $b \in R$ is a zero divisor if there is $a \in R$, $a$ not equal to zero, such that $ab=0$. Also, an element $p$ is irreducible if whenever $p=ab$, either $a$ or $b$ is a unit.
I just don't know how to properly put the two definitions together to formulate a sound proof for this question. I think the best approach would be a proof by contradiction. So assuming $a$ is a zero divisor and then showing that $a$ in R is in fact reducible which leads to a contradiction. But I still don't know how to show that.