I was working through Abstract Algebra : A Geometric Approach (by Theodore Shifrin) and I came across this exercise that I am struggling with.
($ℤ_m$ means integers mod m)
Find All Ring Homomorphisms:
$ϕ:ℤ_m \longrightarrow ℤ_n$ (your answer will depend on the relationship between m and n).
Here is what I have so far:
If $\gcd(m,n) = 1$:
then the only homomorphism is trivial: $ϕ(x) = 0$
If $ m < n$, then:
$ϕ(0) = 0\;$ and $\;ϕ(x) = k$ , such that $k^2 ≡ k\pmod n$ and $2k ≡ 0\pmod n$.
I'm not sure how much of this is correct, or what to do in the case m>n.
Any help would be greatly appreciated!