Let $G$ be a profinite(infinite) group, $\Lambda$ be the complete group ring(Iwasawa algerbra) of $G$ over a unity ring $R$. My first question is that do we know something about the relation with $\Lambda$ and the group ring $R[G]$. I have some feeling that $\Lambda$ contains $R[G]$, but I'm not sure.

My second question is that does $R[G]$ have any arithmetic application, like group rings for finite groups?

The motiviation for this problem is that when I read the book, Selmer complex, JAN NEKOVÁR, I find that in the settings, like on page 95 and 96, there always occur many profinite $G$ and $R[G]$-modules. I'm a little confused about it and not sure whether it is just a quirk of the author to use $R[G]$ to denote $R[[G]]$. Maybe it caused by my ignorance.

But later I find in other article there exist the same expression:

Let $G$ be a profinite group, $R$ a topological ring and let $M$ be a topological $R$-module with a continuous $R$-action and we additionally assume that $M$ is a continuous $R[G]$-module, i.e. the group homomorphism $G \rightarrow Aut_{R,cts}(M)$ is continuous.

So I want to know if it makes sense for using group ring for infinite group to study number theory, what will happen differently with group ring for finite group and complete group ring. Thanks very much.

Let $G$ be a profinite(infinite) group, $\Lambda$ be the complete group ring(Iwasawa algerbra) of $G$ over a unity ring $R$. My first question is that do we know something about the relation with $\Lambda$ and the group ring $R[G]$. I have some feeling that $\Lambda$ contains $R[G]$, but I'm not sure.

My second question is that does $R[G]$ have any arithmetic application, like group rings for finite groups?

The motiviation for this problem is that when I read the book, Selmer complex, JAN NEKOVÁR, I find that in the settings, like on page 95 and 96, there always occur many profinite $G$ and $R[G]$-modules. I'm a little confused about it and not sure whether it is just a quirk of the author to use $R[G]$ to denote $R[[G]]$. Maybe it caused by my ignorance.

But later I find in other article there exists the same expression:

Let $G$ be a profinite group, $R$ a topological ring and let $M$ be a topological $R$-module with a continuous $R$-action and we additionally assume that $M$ is a continuous $R[G]$-module, i.e. the group homomorphism $G \rightarrow Aut_{R,cts}(M)$ is continuous.

So I want to know if it makes sense for using group ring for infinite group to study number theory, what will happen differently with group ring for finite group and complete group ring. Thanks very much.