Let $k[x]$ be the ring of polynomials over a field *k* in one variable *x*. A $k[x]$-module is a *k*-vector space together with a linear endomorphism (the action of *x*).

The field $k(x)$ of rational functions is the maximal localization of $k[x]$, i.e. field of fractions. (Edit: yes, I was an idiot in what I wrote here first; thanks James.)

The ring $k[[x]]$ of formal power series is the completion of $k[x]$ at the ideal $(x)$. It is natural to consider only $k[[x]]$-modules which are likewise *complete*, meaning roughly that we can sum "infinite linear combinations" whose coefficients are increasing powers of *x*. Completeness of a $k[x]$-module automatically makes it a $k[[x]]$-module.

Finally, both $k(x)$ and $k[[x]]$ embed into the ring $k((x))$ of formal Laurent series. I have two questions, which I ask together because they seem related:

Is there a general ring-theoretic construction, akin to localization and completion, which produces $k((x))$ from $k[x]$?

Is there a natural condition to impose on $k((x))$-modules, akin to completeness for $k[[x]]$-modules, which would enable us to sum infinite linear combinations with coefficients increasing in powers of

*x*?