Proposition 4

Under the hypotheses of Definition 5, let $A$ and $m$ (resp. $A'$ and $m'$) be the ring and ideal of the valuation $v$ (resp. $w$*). Then

The ideals of $A'$ containing $mA'$ and distinct from $A'$ correspond to the major subsets of $\Gamma_w$ consisting of elements >0 and containing the elements >0 of $\Gamma_v$ (§ 3, no. 5, Corollary to Proposition 7). They are therefore equal in number to $\varepsilon(w/v)$ and, as they form a totally ordered set under inclusion, this number is equal to the length of the quotient ring $A'/mA'$. Now a module of length 1 over $A'$ is a 1-dimensional vector space over $A'/m'$ and hence a module of length $f(w/v)$ over $A$; hence, as $A'/mA'$ is of length $\varepsilon(w/v)$ over $A'$, it is of length $\varepsilon(w/v) f(w/v)$ over $A$, that is over $A/m$.