# Element of minimum norm among non-units in Euclidean ring is a universal side divisor

## Contents

## Statement

Suppose is a commutative unital ring and is a Euclidean norm on -- in particular, is a Euclidean ring. Suppose is a nonzero element of that is not a unit, and such that for all nonzero non-units of . Then, is a universal side divisor in .

## Definitions used

### Euclidean norm

`Further information: Euclidean norm`

### Universal side divisor

`Further information: Universal side divisor`

A nonzero non-unit in a commutative unital ring is termed a universal side divisor in if for any , either or there exists a unit such that .

## Related facts

- Element of minimum norm in Euclidean ring is a unit
- Every Euclidean ring has a unique smallest Euclidean norm: With respect to this Euclidean norm, the units are precisely the elements of norm zero and the universal side divisors are precisely the elements of norm one.

## Proof

**Given**: A commutative unital ring with a Euclidean norm . A nonzero non-unit element of such that for all nonzero non-units in .

**To prove**: is a universal side divisor in .

**Proof**: Pick any . Then, by the Euclidean algorithm, we can write:

where either or . If , , and we are done. Otherwise, . By assumption, has smallest norm among the non-units, so must be a unit, hence , so for a unit , and we are done.