Thread: Homomorphism ring

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?

Somehow, I suspect the answer to be gcd(n,m), because I read it somewhere.But I actually want to have an idea of how to get there.

Originally Posted by karlito03

Somehow, I suspect the answer to be gcd(n,m), because I read it somewhere.But I actually want to have an idea of how to get there.

Hint:

Spoiler:

If is a homomorphism then the homomorphism is entirely determined by

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)?

if n=m then image of any ring Zn-->Zm is determined by the image of 1 mod (n) where m=n.
now if n>m Zn--> Zm need not to be an homomorphism.

Originally Posted by karlito03

now if n>m Zn--> Zm need not to be an homomorphism.

I'm not sure what you mean here? Why isn't not a homomorphism?

Hint 1: put and extend linearly to all . now look at the conditions that will make well-defined and multiplicative.

Hint 2: using the above, show that is defined by where satisfies the following conditions: and

i dont get the solution for this question. can someone explain it to me? thanks