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!

  • 2
    $\begingroup$ In any of these rings, once you know $f(1)$, you know the entire homomorphism $f$. (That's because $k=\sum_{i=1}^k 1$, so $f(k) = \sum_{i=1}^k f(1)$.) So what are the possibilities for $f(1)$? $\endgroup$ – Robert Shore Apr 9 at 22:02
  • $\begingroup$ wouldn't the possible values for $f(1)$ be $0,...,n-1$ ? I am confused as to how this will help find a generalization $\endgroup$ – Scott B. Apr 9 at 22:17
  • $\begingroup$ Not all of those values are possible because $m=0 \Rightarrow mf(1)=0$. $\endgroup$ – Robert Shore Apr 9 at 22:22
  • $\begingroup$ I don't know what $x$ means in your formula. You should think about two cases: $m<n$ and $m>n$, and divisibility will be crucial. Remember that the definition I take in the text requires a ring homomorphism $\phi\colon R\to S$ to map $1_R$ to $1_S$. $\endgroup$ – Ted Shifrin Apr 13 at 18:21

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