Let $R$ be a [principal ideal domain](/page/Principal%20Ideal%20Domain), viewed as a commutative integral domain with multiplicative identity $1_R$ and zero element $0_R$. For each $x \in R$, let $(x)$ denote the principal ideal generated by $x$, defined by
Then for every ideal $I \trianglelefteq R$, there exists an element $a \in R$ such that $I = (a)$. Moreover, if $a,b \in R$ satisfy $(a) = (b)$, then there exists a unit $u \in R^\times$ such that $a = ub$.