Element of minimum norm in Euclidean ring is a unit
Then, is a unit in .
In particular, all elements of norm zero are units.
- 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.
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.