Let $G$ be an infinite group. It's (integral) group ring $\mathbb{Z}[G]$ has as its elements the *finite* formal linear combinations 
$$
m\_1g\_1 + m\_2g\_2 + \cdots + m\_ng\_n,\qquad n\in\mathbb{N},\quad m\_i\in\mathbb{Z},\quad g\_i\in G,
$$
and these are added and multiplied in the obvious way such that the usual ring axioms are satisfied. Thus it has underlying abelian group isomorphic to a direct sum of copies of $\mathbb{Z}$, with one copy for each group element $g\in G$.


I wonder what happens if we replace *direct sum* with *direct product* in the above construction? Can we make the direct product $\prod\_{g\in G} \mathbb{Z}$ in the category of abelian groups into a ring by simply defining
$$
(\sum m\_gg)(\sum n\_h h) = \sum m\_g n\_h (gh)
$$
and not worrying about whether the sums converge?


And if so, is this ring considered anywhere in the literature? Is it some sort of "completed group ring"?



A version of the group ring using direct product rather than direct sum?
OK, I will make my comment into an answer. The multiplication is not well defined, because of infinite sums :-)


