How to I find all the possible values of $f(1)$ when $f$ is the ring homomorphism from $\mathbb{Z}/6\mathbb{Z}$ to $\mathbb{Z}/42\mathbb{Z}$?
My first thought was to see when they are equal like $1$mod6=$1$mod$42$ etc but then this doesn't give me the correct answer and I am not sure why?
The thing for finding a homomorphism of the underlying additive groups is that $$f(1)+f(1)+f(1)+f(1)+f(1)+f(1)=f(0)=0\, $$ So, $7\,|\,f(1)$ must hold.
To preserve multiplication as well, $f(1)$ must also be an idempotent: $f(1)^2=f(1)$, which further decreases the number possibilities.
If we want a unital ring homomorphism (which is, by the way, the most standard meaning of 'ring homomorphism'), then it means $f(1)=1$ which is impossible in this case because of the first item above.