Principal ideal ring

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In mathematics, a principal ideal ring is a ring $R$ such that every ideal $I$ of $R$ is a principal ideal, i.e. generated by a single element $a$ of $R$ .

[edit] Examples

The ring $\mathbb{Z}$ of integers with its usual operations is a principal ideal ring.
The polynomial ring

$\mathbb{Z} \left[ \sqrt{5} \right] = \mathbb{Z} \oplus \sqrt{5} \mathbb{Z}$

is not a principal ideal ring: there is no single element $r \in \mathbb{Z} \left[ \sqrt{5} \right]$ such that the ideal generated by $r$ equals the ideal generated by the two elements $2$ and $\sqrt{5}$ .