Let $R=\mathbb{Z}\_p[[X]]$ where $\mathbb{Z}\_p$ denotes the $p$-adic integers and $p$ is a prime. Then what is $R\_{(p)}$ $(R$ localised at the ideal $pR)$ $?$



Localisation of $\mathbb{Z}_p[[X]]$ at ideal $(p)$
$R$ is a UFD and its primes up to associates are: $p$, and the irreducible polynomials in $\mathbb{Z}\_p[X]$ that are "distinguished," or "Weierstrass," that is, that are congruent mod $p$ to $X^n$, where $n$ is the degree of the polynomial. See Theorem 1.4 of <http://arxiv.org/pdf/1107.4860v4.pdf>


Thus, $R\_{(p)}$ is gotten by inverting all of these except $p$. Specifically, elements of $R$ are of the form $f/g$, where $f$ is an element of $R$ and $g$ is a distinguished polynomial in $\mathbb{Z}\_p[X]$ (which is a product of irreducible distinguished polynomials). Thus $g = X^n + ph$, where $h$ is a polynomial in $\mathbb{Z}\_p[X]$ of degree less than $n$. 


This all can be arrived at from the $p$-adic Weierstrass preparation theorem. 


