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:

$N(b)\leq N(r)\ \forall \ r\neq 0$.

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

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 $R$ with Euclidean norm $N$, such that $N(b)\leq N(r)$ for all $r\neq 0$.

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.