Length of irreducible factorization is strictly multiplicatively monotone on unique factorization domain

Statement

Suppose $R$ is a unique factorization domain. Define $N:R\setminus \{0\}\to \mathbb {N} _{0}$ as follows:

• $N(u)=0$ for any unit.
• $N(up_{1}p_{2}\dots p_{r})=r$ where $u$ is a unit and $p_{i}$ are irreducible elements in $R$.

Note that this norm is well-defined because unique factorization guarantees that every non-unit can be written as a product of irreducibles and any two such expressions are equivalent up to ordering and associates.

Then, $N$ has the following properties:

• $N(ab)=N(a)+N(b)$ for $ab\neq 0$.
• $N(a)=0$ if and only if $a$ is a unit.
• $N(ab)\geq N(a)$ with equality occurring if and only if $b$ is a unit. (This last fact makes $N$ a strictly multiplicatively monotone norm).