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"?

Vertex Operator Algebras and the Monster. Thus, these direct product analogues might be best seen by pursuing analogies with distributions. – Todd Trimble♦ Jan 18 '13 at 14:05