To show that $7 \Bbb Z$ and $16 \Bbb Z$ are isomorphic as groups but not isomorphic as rings. To show that $7 \Bbb Z$ and $16 \Bbb Z$ are isomorphic as groups but not isomorphic as rings.
I have done the first part but finding difficult to show that they are not isomorphic as rings??

$n \mathbf{Z}$ is generated by $n$, so that a group morphism $n \mathbf{Z} \to G$ is characterized by the value it takes on $n$. Therefore, the group morphism $7 \mathbf{Z} \to 16 \mathbf{Z}$ sending $6$ to $17$ is an isomorphism, and its inverse is defined by $16\mapsto 7$. Now take $f : 7 \mathbf{Z} \to 16 \mathbf{Z}$ a ring morphism. Then $f(7)k = 16$ for some $k$. But the $f(7) = 2^d$ for $0\leq d \leq 4$. Can you conclude from that ?
