PID not implies Euclidean

This article gives the statement and possibly, proof, of a non-implication relation between two commutative unital ring properties. That is, it states that every commutative unital ring satisfying the first commutative unital ring property need not satisfy the second commutative unital ring property
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Statement

There exist principal ideal domains that are not Euclidean.

Proof

The following ring is a principal ideal domain which is not Euclidean:

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

Proof that it is a principal ideal domain

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Proof that it is not a Euclidean domain

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