Exercise A0.V.2.16

Let $R$ be a Euclidean domain with valuation $v$ such that $v(ab) \geq v(b)$ for all non-zero $a,b \in R.$

Proposition A

Any two associate elements $a,b \in R$ have the same valuation.

Proof

Let $a, b \in R$ be non-zero associate elements, and let $au = b$ and $bu' = a.$ In that case, $v(au) = v(b) \geq v(a)$ and $v(bu') = v(a) \geq v(b).$ Hence $v(a) = v(b).$

$\blacksquare$

Proposition B

The units of $B$ have minimum valuation.

Proof

Since (by Proposition A) all units have the same valuation, it is enough to prove that $1$ has minimum valuation. Observe that, for all non-zero $a \in R,$ $v(a) = v(a \cdot 1) \geq v(1).$

$\blacksquare$