Multiplicatively monotone norm is constant on associate classes

From Commalg
Jump to: navigation, search

Statement

Suppose R is a commutative unital ring and N is a norm on R that is multiplicatively monotone: N(ab) \ge \max \{ N(a), N(b) \} whenever a,b \in R are such that ab \ne 0. Then, if a,b \in R are associate elements, we have N(a) = N(b).

Note that the proof does not in fact use the fact that the norm is Euclidean.

Proof

Given: A multiplicatively monotone Euclidean norm N on a commutative unital ring R. Two elements a,b \in R that are associate elements.

To prove: N(a) = N(b).

Proof: By definition of associate elements, there exist elements c,d \in R such that a = bc, b = da. By the definition of multiplicatively monotone, we get:

a = bc \implies N(a) \ge \max \{ N(b), N(c) \} \ge N(b).

Similarly:

b = da \implies N(b) \ge N(a).