Definition:Prime Element of Ring

Definition
Let $R$ be a commutative ring.

An element $p \in R$ is prime if $a$ is non-zero, not a unit, and whenever $p$ divides $a b$, with $a, b \in R$, then either $p$ divides $a$ or $p$ divides $b$.

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