example of PID

Important examples of principal ideal domains:

• The ring of the integers $\mathbb{Z}$.

• The ring of polynomials in one variable over a field, i.e. a ring ofthe form $𝔽[X]$, where $𝔽$ is a field. Note that the ring of polynomials in more than one variable over a field is never a PID.

Both of these examples are actually examples of Euclidean rings, which arealways PIDs. There are, however, more complicated examples of PIDs which arenot Euclidean rings.