Principal ideal domain

Remarks

Every Euclidean domain is a principal ideal domain. So $\mathbb{Z}$, $K[x]$ for a field $K$, and $\mathbb{Z}[i]$ are principal ideal domains. The ring $\mathbb{Z}[x]$ is not a principal ideal domain. Every principal ideal domain is a unique factorization domain. The ring $\mathbb{Z}[\tfrac{1 + \sqrt{-19}}{2}]$ is a principal ideal domain that is not a Euclidean domain.