Element of minimum norm in Euclidean ring is a unit

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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.

Related facts

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.