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I've been asked to find all the ring homomorphisms from $R$ to $S$ which aren't the zero homomorphism in the following cases:

a) $R = \mathbb{Z}_{30}, \ S = \mathbb{Z}_{42} $

b) $R = \mathbb{Z}[X], \ S = \mathbb{Q} $

c) $R = \mathbb{Z}D_{8}, \ S = \mathbb{Z} $

d) $R = \mathbb{C}, \ S = \mathbb{R} $.

So for a) its obvious - I should find all the idempotents $\ne 0$ in $\mathbb{Z_{42}}$ and the homomorphisms are the ones that takes 1 to any of that idempotents.

For d) I think its the same but instead of where the homomorphism take 1 I should look where the homomorphism takes $\mathbb{i}$ and the only idempotent $\ne 0$ in $\mathbb{R}$ is 1.

I'm stuck with b) and c). Could anyone give me some hints?

Thanks in advance.

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Apparently you do not require ring homomorphsisms to map $1_R$ to $1_S$. Since most texts do require that, it would be good t be explicit about this. Also please clarify your notation; $\Bbb Z42$ might be thought ot be the rng of integer multiples of $42$, but I'm not sure you meant that. $\Bbb ZD8$ is not clear to me at all. – Marc van Leeuwen Nov 21 '13 at 14:28
In Z42 I don't know how to make the 42 smaller and in the bottom(same with Z30). – Shai Alkoby Nov 21 '13 at 15:07
Just type \mathbb{Z}_{42} (within the dollars). Use an underscore _ for subscripts, a caret ^ for superscripts. – Marc van Leeuwen Nov 21 '13 at 15:32

I would think about what I need to specify to define a hom. For b) for example, I need to decide where $1$ and $X$ are mapped to. I now ask "Are there any conditions on where they can be mapped?''. What could go wrong?

c) is more interesting. $D_8$ can be generated by two elements, $a,b$ say. What are the possible elements I could map $a$ to?

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