Definition:Prime Element of Ring

Definition
Let $R$ be a commutative ring.

Let $p \in R \setminus \left\{{0}\right\}$ be any non-zero element of $R$.

Then $p$ is prime if $p$ is not a unit and whenever $a, b \in R$ such that $p$ divides $a b$, then either $p$ divides $a$ or $p$ divides $b$.

Also see

 * Prime Element iff Generates Prime Ideal, where it is shown that this is equivalent to the statement that the ideal $(p)$ generated by $p$ is prime.