I encountered the following passage in Matsumura's *Commutative Ring Theory* :

A a Noetherian ring, $B=A[[x]]$ a formal power series ring. $M\subset B$ a maximal ideal, $\mathfrak{m}=M\cap A$. Then $(B_{M})^{\mbox{^}}=(A_{\mathfrak{m}})\mbox{^}[[x]]$, where ^ indicate $M$-adic and $\mathfrak{m}$-adic completions, respectively.

It's not immediately clear to me why this is the case. How should I go about proving this? Thanks!

completionof the localization at amaximalideal, there is no need to invert anything new: one just forms the inverse limit of quotients $B/M^n$ (and similarly for $A$). Here we use the evident fact that $B/M^n \rightarrow B_M/(M B_M)^n$ is an isomorphism for maximal $M$ (and similarly for $A$) due to localization commuting with the formation of quotients. So you don't need to invert anything new when doing the calculation, regardless of whether $A$ or $B$ is local. – Boyarsky Jun 25 '10 at 10:01