prime element

An element $p$ in a ring $R$ is a prime element if it generates a prime ideal. If $R$ is commutative, this is equivalent to saying that for all $a,b\in R$ , if $p$ divides $ab$, then $p$ divides $a$ or $p$ divides $b$.

When $R=\mathbb{Z}$ the prime elements as formulated above are simply prime numbers.