\begin{definition}[Definition:Prime Element of Ring]
Let $R$ be a commutative ring.
Let $p \in R \setminus \set 0$ be any non-zero element of $R$.
Then $p$ is a '''prime element of $R$''' {{iff}}:
:$(1): \quad p$ is not a unit of $R$ 
:$(2): \quad$ whenever $a, b \in R$ such that $p$ divides $a b$, then either $p$ divides $a$ or $p$ divides $b$.
\end{definition}