Reid Barton's very nice answer to Computing the structure of the group completion of an abelian monoid, how hard can it be?Computing the structure of the group completion of an abelian monoid, how hard can it be? contained a pointer to the wikipedia page for the Eilenberg-Mazur swindle. Towards the bottom of that page, one finds the following relevant paragraph.
Example: (Lam 2003, Exercise 8.16) If $A$ and $B$ are any groups then the Eilenberg swindle can be used to construct a ring $R$ such that the group rings $R[A]$ and $R[B]$ are isomorphic rings: take $R$ to be the group ring of $A + B + A + B + \ldots$
