Completion of a metric space Let $R$ be a topological ring (i.e addition and product are continuous) which we assume it is metrizable with metric d and consider the completion $\hat{R}$ of the ring $R$ defined as the set of all classes of equivalent Cauchy sequences. Question: why is the completion of $\hat{R}$ (defined in this way) also a topological ring?
It is certainly metrizable by the standard metric: $\hat{R}$, $\hat{d}([(x_{n})],[(y_{n})])=\operatorname{lim} d(x_{n},y_{n})$.
To check addition is continuous: suppose $([(x_{n})],[(y_{n})])$ is a sequence of elements in $\hat{R} \times \hat{R}$ such that$([(x_{n})],[(y_{n})]) \rightarrow ([x],[y])$ then by by definition of convergence in a metric product space we have that:
$[(x_{n})] \rightarrow [x]$ and $[(y_{n})] \rightarrow y$
Therefore $d(x_n,x) \rightarrow 0$ and $d(y_n,y) \rightarrow 0$ as $n \rightarrow \infty$. Since we are in metric spaces this says $x_{n} \rightarrow x$ and $y_{n} \rightarrow y$ as $n \rightarrow \infty$. By assumption $R  \times R$ is continuous with respect addition so $x_{n} + y_{n} \rightarrow x+y$.
But then $\hat{d}([(x_{n}+y_{n})])=[x+y]$ so we have that addition is continuous. 
Is this OK?
 A: This is false if you don't posit some additional relationship between the metric and the structure maps of the ring beyond that they are continuous. Consider $\mathbb{R}$ equipped with the metric
$$d(x, y) = |\arctan x - \arctan y|$$
(where $\arctan x \in (-\frac{\pi}{2}, \frac{\pi}{2})$). This induces the same topology as the usual metric so addition and multiplication are still continuous, but the completion with respect to this metric is the extended real line $\mathbb{R} \cup \{ +\infty, -\infty \}$ and it is not possible to extend either the addition or the multiplication.
Your argument fails here:

By assumption $R \times R$ is continuous with respect addition so $x_n + y_n \to x + y$.

$x + y$ is not well-defined at this step in the argument if either $x$ or $y$ does not lie in $R$. What you probably meant is something like this: the sum of Cauchy sequences is Cauchy, so we can define $x + y$ by taking $\lim (x_n + y_n)$.
Unfortunately, the continuity of addition does not guarantee that the sum of Cauchy sequences is Cauchy; in the above example, take
$$x_n = (-1)^n + n, y_n = (-1)^n - n.$$
Even when the sum of two Cauchy sequences is Cauchy, it's false that $\lim (x_n + y_n)$ is uniquely determined by $\lim x_n$ and $\lim y_n$; to see this, take
$$x_n = -n, y_n = n, x_n = -n, y_n = n^2.$$
Edit: for the $I$-adic completion things are considerably simpler because we can write $d(x, y) = |x - y|$ for some function $| \cdot | : A \to \mathbb{R}$ satisfying (among other things) $|x + y| \le |x| + |y|$ and $|xy| \le |x| |y|$. These properties readily imply all of the nice properties that you want. 
In my opinion the conceptually cleanest way to think about $I$-adic completion is not using Cauchy sequences but as follows: the $I$-adic completion is the inverse limit of the quotients $R/I^n$. 
