Principal ideal domain

A principal ideal domain (PID) is an integral domain $R$ such that every ideal $I\subseteq R$ is principal , i.e. $I=(a)$ for some $a\in R$.

PIDs provide strong control of divisibility and modules, and they sit between Euclidean domains and unique factorization: every Euclidean domain is a PID, and every PID is a UFD (see PID implies UFD ).

Examples:

• $\mathbb{Z}$ is a PID.
• If $k$ is a field, then $k[x]$ is a PID (indeed Euclidean).
• $k[x,y]$ is not a PID since $(x,y)$ is not principal.