Consider the ring $\Bbb{Q}[\varepsilon]:=\Bbb{Q}[X]/(X^2)$ and its natural subrings $\Bbb{Z}[\varepsilon]$ and $\Bbb{Q}$ as abelian groups. Then by the second isomorphism theorem $$\Bbb{Z}[\varepsilon]/(\Bbb{Z}[\varepsilon]\cap\Bbb{Q}) \cong(\Bbb{Z}[\varepsilon]\cdot\Bbb{Q})/\Bbb{Q},$$ But the left hand side is isomorphic to $\Bbb{Z}$ whereas the right hand side is isomorphic to $\Bbb{Q}$. Where is my misunderstanding?
EDIT: The second isomorphism theorem states that for a group $G$, a subgroup $S$ and a normal subgroup $N\subset G$ there is a canonical isomorphism $$S/(S\cap N)\cong(SN)/N,$$ where $SN$ is the subgroup of $G$ generated by $S$ and $N$.