Example

The ring $\mathbf{Z}$ of rational integers is a principal ideal domain (I, p. 111). If $K$ is a commutative field, then the polynomial ring $K[X]$ in one indeterminate over $K$ is a principal ideal domain (IV, p. 11, Prop. 11); the same is true of the ring of formal power series $K[[X]]$, for every ideal of this ring has the form $(X^n)$ (IV, p. 38, Prop. 12). \* The ring of integers in a p-adic field is a principal ideal domain. \*

If $\mathbf{Q}(i)$ denotes the field obtained from the field $\mathbf{Q}$ of rational numbers by adjoining a root $i$ of the irreducible polynomial $X^2 + 1$, then the elements $a + bi$ of $\mathbf{Q}(i)$, where $a$ and $b$ are rational integers, form a subring $\mathbf{A}$ of $\mathbf{Q}(i)$, called « the Gaussian integers », which is a principal ideal domain (VII, p. 50, Ex. 7). By contrast, in the field $\mathbf{Q}(\rho)$, where $\rho$ is a root of $X^2 + 5$, the subring $B$ consisting of elements $a + bp$ (a and b rational integers) is not a principal ideal domain (VII, p. 51, Ex. 12).

The polynomial ring $K[X, Y]$ in two indeterminates over a field $K$ is not a principal ideal domain. Indeed the nonzero constants are the only elements which divide both $X$ and $Y$, and none of them generates the ideal generated by $X$ and $Y$.