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 a strictly multiplicatively monotone norm 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.