Show that there are at most $4$ ring homomorphisms between $\Bbb Z[S_3] \to \Bbb Z[\Bbb Z \backslash 2 \Bbb Z]$.
Here is what I did :
We know that ring homomorphisms send inversible elements to inversible elements. Since $1.(123)$ is inversible then $f(1.(123))$ is inversible in $\Bbb Z[\Bbb Z \backslash 2 \Bbb Z]$. Inversible elements in $\Bbb Z[\Bbb Z \backslash 2 \Bbb Z]$ are $±1.\overline{0}$ and $±1.\overline{1}$ so at most $4$ possible images for $1.(123)$ so at most $4$ ring homomorphisms between $\Bbb Z[S_3] \to \Bbb Z[\Bbb Z \backslash 2 \Bbb Z]$.
What should I do next ? I would like to say that when an image for $1.(123)$ is chosen then all others elements are fixed but I struggle a bit.