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.
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