Euclidean ring that is not a field has a universal side divisor

Statement
Suppose $$R$$ is a fact about::Euclidean ring (for instance, a fact about::Euclidean domain) that is not a field. Then, $$R$$ has a universal side divisor.

Facts used

 * 1) uses::Element of minimum norm among non-units in Euclidean ring is a universal side divisor

Proof
In a Euclidean ring that is not a field, there exist nonzero elements that are not units. Since the norm function goes to a well-ordered set, there must exist an element of minimum norm among these. By fact (1), such an element is a universal side divisor, and we are done.