So I'm attempting exercise 30 of chapter 5 of Atiyah-Macdonald. There we have the following definition:
Let $A$ be a valuation ring of a field $K$. The group $U$ of units of $A$ is a subgroup of the multiplicative group $K^{*}$ OF $K$. Let $\Gamma = K^{*}/U$. If $\xi, \eta \in \Gamma$, are 'represented' by $x, y \in A$, define $\xi \geq \eta$ to mean $xy^{-1}\in A$.
I am confused by this definition. So I am trying to show this defines a total order. And I was wondering what it means for "$\eta = \xi$", in particular, what do they mean by 'represented' by $x, y \in A$.
Any insights or help is appreciated.
Cheers !
Since $U\subseteq K^*$ is a subgroup, then $\eta, \xi$ are cosets of $U$ in $K^*$. Write $\eta = yU$ and $\xi=xU$ for some $x,y\in K^*$. We say that $\eta, \xi$ are represented by $y,x$. Then I think Atiyah-Macdonald define
$$\xi \geq \eta\,\Leftrightarrow xy^{-1}\in A$$
Of course one need to verify that this definition is correct.