This question does not let me sleep.
Fina all Ring homomorphism between Z_n----> Z_m.
I know there should be 3 case:
n>m
n<m and m=n.
the last one is trivial. Anyways, how do you guys think I should proceed or what should I consider?
I sure ..
call theta (f).
Then f(1)=1 since you assume it was a ring homomorphism. But how can we get to the point of saying that all homomorphism must be determined by f(1). There must be some cases. For instance m>n, n>m or m=n or Don't m and n have to be such that (n,m)=1. How many homomorphism can one get in each case. Are you saying they will all be determined by f(1)?
Hint 1: put $\displaystyle \theta(1+n \mathbb{Z})=r+m\mathbb{Z}$ and extend $\displaystyle \theta$ linearly to all $\displaystyle \mathbb{Z}/n\mathbb{Z}$. now look at the conditions that will make $\displaystyle \theta$ well-defined and multiplicative.
Hint 2: using the above, show that $\displaystyle \theta$ is defined by $\displaystyle \theta(x+n\mathbb{Z})=rx + m\mathbb{Z},$ where $\displaystyle r$ satisfies the following conditions: $\displaystyle \frac{m}{\gcd(n,m)} \mid r$ and $\displaystyle m \mid r^2-r.$