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I'm seeing a bunch of questions such as "find all ring homomorphisms from $\mathbb{Z}$ to $\mathbb{Z}/30\mathbb{Z}$," etc., lately, so I'm wondering about whether there's a general method for doing this between any two rings. What steps should I take? I'd appreciate an example in your answer, if you can manage one. Thanks so much!

EDIT: I should be clear here that I'd not necessarily like to see methods that work in every case, and I don't necessarily need a lot of rigour either. I'd just like to see what works for you in most practical cases.

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    $\begingroup$ I don't see how to approach this in all generality. But in case of a ringhomomorphismen $\phi: \Bbb Z \rightarrow R$, there isn't any choice. $1_{\Bbb Z}$ maps to $1_R$ and this determines $\phi$ completely. $\endgroup$ – Stefan Mesken Jun 19 '14 at 0:56
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    $\begingroup$ It is important to note that, for all ring homomorphisms $\phi:R \rightarrow S$, $\phi(1_R) = 1_S$. Oftentimes, this will uniquely determine the homomorphism! For example, this makes it easy to see that there is only one homomorphism $\mathbb{Z} \rightarrow \mathbb{Z}$. $\endgroup$ – Kaj Hansen Jun 19 '14 at 0:56
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    $\begingroup$ There is no general method, simply because there are way too many rings. $\endgroup$ – Mariano Suárez-Álvarez Jun 19 '14 at 0:57
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    $\begingroup$ In the simpler versions of this question you should carefully apply all of the ring axioms and just see what happens. $\endgroup$ – Qiaochu Yuan Jun 19 '14 at 1:42
  • $\begingroup$ I think you guys misunderstand the context. I'm not going for ultra-rigorous proofs-and-theorems methods here. I'd just like to see some techniques that work in common cases. For example, what's a good first step to take? I really don't know if I'm at the level where I'd be seeing any crazy exceptions anyway, so yeah. Rules of thumb. I'll edit my question to make this more apparent. $\endgroup$ – Samuel Yusim Jun 19 '14 at 21:40

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