I would like to understand if there is a connection between these definitions of completion:
1. Completion of a ring with respect to an ideal (which I found in the German Wikipedia article)
Let $R$ be a ring, and $I \subseteq R$ be an ideal. Then $R^{\Bbb{N}}$ is ring of sequences in $R$. We call a sequence $(a_i) \in R^{\Bbb{N}}$ a Cauchy sequence if
$$
\forall n \in \Bbb{N} \: \exists k \in \Bbb{N} \: \forall i,j > k \: : \: a_i-a_j \in I^n,
$$
and call it a zero sequence if
$$
\forall n \in \Bbb{N} \: \exists k \in \Bbb{N} \: \forall i > k \: : \: a_i \in I^n.
$$
The set $\textbf{CF}$ of all Cauchy sequences in $R$ constitute a subring in $R^\Bbb{N}$, and the set $\textbf{NF}$ of all zero sequences in $R$ is an ideal of $\textbf{CF}$. Therefore, we can define
$$
\hat{R}_I := \textbf{CF}/\textbf{NF}
$$
which is called the completion of $R$ with respect to $I$.
2. Completions of rings as an inverse limit (also taken from the cited Wikipedia article above)
Now it can be shown that
$$
\hat{R}_I \simeq \varprojlim_n (R_n,f_n)_{n \in \Bbb{N}} := \left\{ (x_n) \in \prod_n R_n \: \middle| \: x_n \in R_n, \: f_n(x_n) = x_{n-1} \right\}
$$
where $R_n = R/I^n$ and $f_n : R_n \to R_{n-1}$, $\bar{x} \mapsto \bar{x}$ (in their respective equivalence classes, of course).
An example I know for that are the $p$-adic integers $\Bbb{Z}_p$ which is the completion of $R = \Bbb{Z}$ wrt. the ideal $I = (p)$, i.e. $\hat{\Bbb{Z}}_{(p)} = \Bbb{Z}_p$.
3. Completions of valued fields
Let $(K,|\cdot|)$ be a valued field, i.e. a field $K$ with an absolute value $|\cdot|: K \to \Bbb{R}$. Here, we can define Cauchy sequences and zero sequences in the usual sense for metric spaces. Also here, we can take the quotient $\tilde{K}$ of all Cauchy sequences in $K$ and all zero sequences in $K$, and call it the completion of $K$ wrt. $| \cdot |$.
I am also familiar with one particular example: For instance, I know that if we take $K=\mathbb{Q}$, and denote $|\cdot | = | \cdot |_p$ to be the $p$-adic absolute value on $K$, the completion of $K$ wrt. $|\cdot |_p$ would be $\tilde{K} = \mathbb{Q}_p$.
What do I want to know?
Now according to the discussion in this post of mine (in the comments of the answer and which I have a hard time understanding), there seems to be a connection between these concepts. I am more familiar with 3., while I found 1. and 2. just recently.
In particular, I am interested if there is a connection from 1./2. to 3.. In 1./2., there is only a ring while in 3., we have a field (where all ideals are the zero ideal or the field itself). Also, the definition is 1./2. in quite algebraic while the one in 3. is rather analytical.
To make this more concrete: Let $R$ be an integral domain and $I \subseteq R$ be an ideal. Let $K = Q(R)$ be the field of fractions of $R$ where I can define a valuation $|\cdot|: K \to \Bbb{R}$ to be $|x| := e^{-v(x)}$ where $v(a) = n \Leftrightarrow a \in I^n \setminus I^{n+1}$ (defined over $R$) and $v\left( \frac{a}{b}\right) = v(a) – v(b)$ (where $a,b \in R$).
Question In the setting I just described, do we have $Q\left( \hat{R}_I \right) = \tilde{K}$?
Best Answer
Yes it is true.in fact you can do better than that: take any $w\in I$, then $\tilde K=\tilde R[\frac{1}{w}]$.(I assume that you really have a valuation field which means that $R$ is local and $I$ is principal and maximal, otherwise you don't get a valuation but see the third paragraph for more general setting)
this is true because you can easily see that components of a Cauchy sequence($a_k$) in $K$ have bounded valuations(the whole sequence is bounded) hence there is a power $n$ of $W$ such that $(w^n a_k)=(b_k)$ is a cauchy sequence in $R$ and hence an element of $\tilde R$. in this way you get a map $\tilde K\to \tilde R[\frac{1}{w}]$ and it is not hard to see that this is an isomorphism.
in fact if you are interested there is a more general theory of this things called tate rings: a Tate ring is a topological ring $R$ and a subring $R_0$ and an ideal $w\in I$ of $R_0$ such that $R=R_0[\frac{1}{w}]$ and the base for the topology is given by $I^n R_0$. in your language $R=K,R_0=R,I=I$ is an example of a tate ring. for a tate you can always define $\tilde{R}=\tilde{R_0}[\frac{1}{w}]$