Principal ideal domain admits multiplicative Dedekind-Hasse norm

Statement
Suppose $$R$$ is a fact about::principal ideal domain. Then, $$R$$ admits a fact about::Dedekind-Hasse norm. In fact, the norm can be modified to a fact about::multiplicative Dedekind-Hasse norm.

Facts used

 * 1) uses::PID implies UFD
 * 2) uses::PID implies Bezout
 * 3) uses::Length of irreducible factorization is strictly multiplicatively monotone on unique factorization domain
 * 4) uses::Strictly multiplicatively monotone norm on Bezout domain is a Dedekind-Hasse norm

Proof
Given: A principal ideal domain $$R$$.

To prove: $$R$$ admits a multiplicative Dedekind-Hasse norm.

Proof:


 * 1) By fact (1), $$R$$ is a unique factorization domain, so by fact (3), the length of irreducible factorization, say $$L$$, defines a strictly multiplicatively monotone norm on $$R$$.
 * 2) By fact (2), $$R$$ is a Bezout domain, so by fact (4), $$L$$ is a Dedekind-Hasse norm on $$R$$.
 * 3) Note that $$L$$ is not multiplicative. However, we do have $$L(ab) = L(a) + L(b)$$. Thus, we can consider the norm $$N(x) = 2^{L(x)}$$ to obtain a multiplicative Dedekind-Hasse norm.