Looking for a definition of R[[G]], i.e. formal power series on groups

Question

I am looking for a definition of a ring of formal power series, just like $R[[X]]$, where $R$ is a commutative ring, but with another group $G$ built in. Let $G$ be a group. In other words it is not clear to me what the transition

$R[G]\rightsquigarrow R[[G]]$

from group rings to "something" would be, which should be analogous to the transition from the ring of polynomials to the ring of formal power series

$R[X]\rightsquigarrow R[[X]]$.

In fact it would be enough for me to know such a definition for $R=\mathbb{Z}$.

Answer 1

Let $R$ be a commutative ring and let $G$ be a group (or more generally a monoid). There is a thing that deserves to be called a completed group algebra $R[[G]]$, although it may not be quite what you expected. As Espen says, in general given a ring $S$ and a two-sided ideal $I$ of that ring there is an $I$-adic completion given by the cofiltered limit of the quotients $S/I^n$. When $S = R[x_1, \dots x_n]$ is a polynomial ring we can take $I = (x_1, \dots x_n)$ and the resulting completion is a formal power series ring.

When $S = R[G]$ is a group algebra there is a different distinguished ideal to care about, namely the augmentation ideal, generated by the elements $g - 1$ for all $g \in G$. The idea intuitively is to think of all of the elements of $G$ as being close to the identity, and hence to think of all of the elements $g - 1$ as being small. Among other things, if $R$ contains $\mathbb{Q}$ then this allows us to make sense of the formal logarithm

$$\log g = \sum_{n \ge 1} \frac{(g - 1)^n}{n}$$

of every $g \in G$.

This is one step in the construction of the Malcev completion. Note that if we set $G = \mathbb{Z}_{\ge 0}$, so that $R[G] \cong R[x]$, then $R[[G]]$ is not $R[[x]]$; rather, it is $R[[x - 1]]$.

Answer 2

Assuming you want a version of $R[G]$ where you take infinite sums instead of finite ones, there is no generalization in general, for the simple reason that infinte sums in rings are not defined in general. The reason the transition works in the classical case of $\mathbb{Z}_{\geq 0}$, the monoid of nonnegative integers, is that there is only finitely many distinct decompositions of a given nonnegative integer into a sum of two nonnegative integers. This is not the case for an arbitrary infinite monoid/group.

Note that even $R[[\mathbb{Z}]]$ is undefined, and $R[[G]]$ is just $R[G]$ for a finite group, so really you are interested in monoids here. The completion $R[X]\rightarrow R[[X]]$ is given with respect to the filtration by the ideal $\langle X\rangle$. If your monoid has a similar filtration by an ideal $I$, then you could do a similar completion $R[M]\rightarrow R[[M]]$, but the result will depend on which ideal you complete with respect to.