# Element of minimum norm in Euclidean ring is a unit

## Statement

Suppose $R$ is a Euclidean ring, i.e., $R$ is a commutative unital ring with a Euclidean norm $N$. Suppose $b \in R$ is a nonzero element such that:

$N(b) \le N(r) \ \forall \ r \ne 0$.

Then, $b$ is a unit in $R$.

In particular, all elements of norm zero are units.

## Proof

Given: A commutative unital ring $R$ with Euclidean norm $N$, such that $N(b) \le N(r)$ for all $r \ne 0$.

To prove: $b$ is a unit in $R$.

Proof: By the Euclidean algorithm, we can write:

$1 = bq + r, \qquad q,r \in R$

where $r = 0$ or $N(r) < N(b)$. By the assumption, $N(r) < N(b)$ is impossible, so we are forced to have $r = 0$. Thus:

$1 = bq$,

and we obtain that $b$ is a unit.