Given that $R$ is a subring of a commutative ring $S,$ and the additive group $S/R$ is of finite order $n$. If $(m,n)=1,$ I wanted to show that $R/mR$ and $S/mS$ are isomorphic rings.
I know that to show $R/mR≅S/mS,$ we need to write down a well-defined map between the two rings and show it is an isomorphism. My problem is how to define the map! Can anyone guide me? Thanks.