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"?
