I have multiples questions actually. Let's introduce the context. We consider $R$ a Dedekind ring, and the valuations $v_\mathfrak{p}$ associated with every $\mathfrak{p}$ prime ideal over $R$.
In Local fields, Serre, there is written some formulas which are "immediate", among others this one : $v_\mathfrak{p}(\mathfrak{a}+\mathfrak{b}) = Inf(v_{\mathfrak{p}}(a), v_{\mathfrak{p}}(b))$.
Actually, I don't understand why. The formula is clearly true if $v_{\mathfrak{p}}(a) \neq v_{\mathfrak{p}}(a)$, but otherwise nothing indicates the equality above ?
Then, given $(x_1, \dots , x_r) \in R^n, (n_1, \dots n_r) \in \mathbb{Z}$, and $\mathfrak{p}_1, \dots, \mathfrak{p}_r$ prime ideal of $R$, we can find $x \in K$ such that : $v_{\mathfrak{p}_i}(x-x_i) \geq n_i$, cause this is a consequence of the theorem of Weak approximation.
But now, we want two things :
1) Find such a $x$ but with the condition that $v_\mathfrak{q}(x) \geq 0$ for all $\mathfrak{q} \neq \mathfrak{p}_1, \dots \mathfrak{p}_n$ prime ideal.
2) In the case where $R$ has only a finite numbers of prime ideal, showing that $R$ is principal. In Local fields, Serre, there is written : "if $\mathfrak{p}$ is a prime ideal, there exists $x \in A$ with $v_{\mathfrak{p}}(x)=1$ and $v_{\mathfrak{q}}(x)=0$ if $\mathfrak{p} \neq \mathfrak{q}$.
So, if $v_\mathfrak{p}(\mathfrak{a}+\mathfrak{b}) = \inf(v_{\mathfrak{p}}(a), v_{\mathfrak{p}}(b))$ was true, I have proven this, but as I think it's a wrong assertion because I think it's rather $\geq$ than $=$ in the formula, then I didn't succeed to prove the points 1) and 2).
Last question, I saw that we can do the same thing but with the condition : $v_{\mathfrak{p}_i}(x-x_i) \geq n_i$ replaced with $v_{\mathfrak{p}_i}(x-x_i) = n_i$.
How to do that ? I tried hundred things, and nothing worked... I think I really need help on this, please !
Thank you !
Given $x \in Frac(R)$ and finitely many primes $P_1,\ldots,P_J$ where $v_{P_j}(x)\ge 0$,
For any $n$ take $$a_{j,n}\in P_j^n,b_{j,n}\in \prod_{i \ne j} P_i^n, a_{j,n}+b_{j,n}=1,\qquad c_{j,n} \in R, v_{P_j}(x-c_{j,n}) \ge n$$ Let $$x_n = \sum_j c_{j,n} b_{j,n}\in R $$ $$\begin{eqnarray} v_{P_j}(x-x_n)&\ge& \min( v_{P_j}( x-c_{j,n} b_{j,n}),\min_{i\ne j} v_{P_j}( c_{i,n} b_{i,n}))\\ &\ge& \min( v_{P_j}( 1-b_{j,n}),\min_{i\ne j} v_{P_j}(b_{i,n}))\\ & \ge & \min(v_{P_j}(a_{j,n}),n) \\ & \ge & n\end{eqnarray}$$