Principal ideal domain admits multiplicative Dedekind-Hasse norm

Statement

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

Facts used

1. PID implies UFD
2. PID implies Bezout
3. Length of irreducible factorization is strictly multiplicatively monotone on unique factorization domain
4. 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.