This is a pretty basic question, but I suspect it might be too exotic for math.stackexchange.

Let $\mathbb{Z}_p$ be the $p$-adic integers. For free pro-$p$ group $F_r$ of rank $r$, we can consider the complete group algebra

$$\mathbb{Z}_p[[F_r]] := \varprojlim_U \mathbb{Z}_p[F_r/U]$$ as $U$ ranges over all open subgroups of $F_r$. Let $x_1,\ldots,x_r$ are generators for $F_r$, which we identify with their images in $\mathbb{Z}_p[[F_r]]$. We may consider the ring of formal power series $\mathbb{Z}_p[[u_1,\ldots,u_r]]$ in the noncommuting variables $u_1,\ldots,u_r$.

According to various sources (Ribes-Zalesski 5.9.1, Ihara - On Galois Representations Arising from Towers...), by sending $u_i\mapsto (x_i-1)$, apparently we get an isomorphism $$\mathbb{Z}_p[[u_1,\ldots,u_r]]_\text{nc}\stackrel{\sim}{\longrightarrow} \mathbb{Z}_p[[F_r]]$$ I'm trying to work out the simplest case, where $r = 1$, where I think of $F_1$ as $\varprojlim_n \mu_{p^n}$ with generator $x = (\zeta_p,\zeta_{p^2},\zeta_{p^3},\ldots)$ and I don't understand how the image of, say, $$a_0 + a_1u + a_2u^2 + a_3u^3 + \cdots$$ is well defined, where $u = u_1$. Even if each $a_i = 1$, the image of $1+u+u^2 + \cdots$ in the group ring $\mathbb{Z}_p[\mu_p]$ should be $$1 + (\zeta_p-1) + (\zeta_p-1)^2 + (\zeta_p-1)^3 + \cdots$$ which would seem to have constant term ``$1-1+1-1+1-1+1-\cdots$''.

Okay, so this is probably linked to the remark that the augmentation ideal is topologically nilpotent, which I don't see. Is it still generated by $f-1$, $f\in F_1$? (perhaps $x-1,x^2-1,x^3-1,\ldots$ would do?). Even if it's some weird topological thing I'm missing, surely that shouldn't come into play at the "finite level" $\mathbb{Z}_p[\mu_p]$?

Not sure what the best tags are for this...