This conjecture (over integeral group rings) states $$\Bbb{Z}G\cong \Bbb{Z}H \implies G\cong H$$.
It can very well be studied over various commutative rings and people do that fixing some commutative ring and finding all classes of group that can be determined by that ring. But do we also study it over noncommutative rings, as I have never seen it any book or paper etc.
I am asking as why are not people interested in fixing a particular noncommutative ring and then finding classes of group which are determined by that $R$, or if some results are known in this direction please lemme know.