# Element of minimum norm in Euclidean ring is a unit

From Commalg

## Statement

Suppose is a Euclidean ring, i.e., is a commutative unital ring with a Euclidean norm . Suppose is a nonzero element such that:

.

Then, is a unit in .

In particular, all elements of norm zero are units.

## Related facts

- Unit in Euclidean ring need not have minimum norm
- Element of minimum norm among non-units in Euclidean ring is a universal side divisor
- Every Euclidean ring has a unique smallest Euclidean norm: With respect to this Euclidean norm, the units are precisely the elements of norm zero.

## Proof

**Given**: A commutative unital ring with Euclidean norm , such that for all .

**To prove**: is a unit in .

**Proof**: By the Euclidean algorithm, we can write:

where or . By the assumption, is impossible, so we are forced to have . Thus:

,

and we obtain that is a unit.