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 localization of $k[x]$ at the multiplicative set of multiples of x. A $k(x)$-module is a $k[x]$-module for which the action of x is invertible.

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:

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

2. 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?

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:

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

2. 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?