PID not implies Euclidean

Statement
There exist principal ideal domains that are not Euclidean.

Facts used

 * 1) uses::Euclidean ring that is not a field has a universal side divisor
 * 2) uses::PID need not have a universal side divisor

Proof
The proof follows from facts (1) and (2). A specific example of a PID that does not have a universal side divisor is:

$$\mathbb{Z}\left[\frac{1 + \sqrt{-19}}{2}\right]$$

Journal references

 * A Principal Ideal Ring that is not a Euclidean ring by Jack C. Wilson, Math. Mag., pp.34-38