Normalized valuations of $A$ (corresponding to the prime elements of $A$).

Question

So, while preparing my Algebra final, i found a paper about a generalization of the idea of an euclidean ring, and thought that reading it would be enriching. However, in certain corollary (corollary 2, section 2), the concept normalized valuations of $A$ (corresponding to the prime elements of $A$) is used. I do not have the slightest idea of what this means. Any help would be appreciated. In case the concept is too complex to be explained shortly, a reference to look this definition up would still be really useful. The paper is https://core.ac.uk/download/pdf/82126785.pdf. (Edit: I understand that probably, with my little knowledge, this kind of documents are something still out of my reach, but I thought it would still be an interesting experience).

Answer 1

Let $p$ be a prime element of a factorial domain $A$ (note that euclidean domains are factorial). Then every element $x\neq 0$ of the field of fractions $K$ of $A$ can be written as

$x=\frac{a}{b}p^e$,

where $a,b\in A$ are not divisible by $p$ and $e\in\mathbb{Z}$. If one requires $a$ and $b$ to possess no common divisor, this representation is unique up to multiplication by units of $A$.

The normalized valuation corresponding to $p$ is the map

$v_p: K\setminus\{0\}\rightarrow\mathbb{Z}, x\mapsto e.$