Definition 10.120.1. Let $R$ be a domain.
Elements $x, y \in R$ are called associates if there exists a unit $u \in R^*$ such that $x = uy$.
An element $x \in R$ is called irreducible if it is nonzero, not a unit and whenever $x = yz$, $y, z \in R$, then $y$ is either a unit or an associate of $x$.
An element $x \in R$ is called prime if the ideal generated by $x$ is a prime ideal.