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

Statement

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

Definitions used

Euclidean ring, Euclidean norm

Further information: Euclidean norm

Universal side divisor

Further information: Universal side divisor

Facts used

1. 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.

References

Textbook references

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