Difference between the completed group algebra and the profinite completion of a group ring

Question

Let $G$ be a reasonably nice group, say residually finite if need be.

We may consider the group algebra $\mathbb{Z}[G]$.

Let $\widehat{\mathbb{Z}[G]} := \varprojlim_I\mathbb{Z}[G]/I$ be the profinite completion, where $I$ runs over all ideals of finite index.

Let $\widehat{\mathbb{Z}}[[\widehat{G}]] := \varprojlim_{n,U}(\mathbb{Z}/n)[G/U]$ be the "completed group algebra", where $n$ runs over all positive integers, and $U$ all finite index normal subgroups of $G$.

When are these two objects the same? (Are they always the same? Perhaps they're the same when $G$ is abelian?)

This seems like a natural question, but I cannot find it addressed anywhere, and I can't seem to deduce anything nontrivial in either direction.

Answer 1

It seems to me that these are indeed isomorphic. Namely, if $n$ is an integer and $U$ is a finite index normal subgroup of $G$, then the kernel $I_{n,U}$ of the natural map $\mathbb ZG\to \mathbb Z/n[G/U]$ is an ideal of $\mathbb ZG$ of finite index and hence part of the inverse system defining $\widehat{\mathbb ZG}$. On the other hand, if $I$ is a finite index ideal of $\mathbb ZG$, then $R=\mathbb ZG/I$ is a finite ring of some characteristic $n$ and the image of $G$ under the natural map is finite hence of the form $G/U$ with $U$ a finite index normal subgroup of $G$. Thus $I_{n,U}$ is contained in $I$. Therefore, the collection of ideals $I_{n,U}$ form a cofinal system of finite index ideals and hence the two completions are isomorphic by standard properties of inverse limits.