I want to find all rings homomorphisms from:
i) $$\mathbb{Z} \rightarrow \mathbb{Z}_m$$
ii) $$\mathbb{Z}_m\rightarrow \mathbb{Z}$$
iii) $$\mathbb{Z}_n \rightarrow \mathbb{Z}_m$$
I don't know how to work this exercise, can someone explain to me how we think about these types of questions?
For the first one I only found $2$ the trivial and $\varphi(m)=m \pmod{n} $
for ii) I think of the trivial and $\varphi(m)=m $
and for iii) I found the trivial and $\varphi(m)=am, a\in \mathbb{Z}_m$ such that $na=0 \pmod{m}$
Assumption: ring homomorphisms send 1 to 1.
The only possible ring homomorphism of type i) is $\varphi(n) = \overline{n}$ - the canonical projection, for $\mathbb{Z}$ is the initial object in the category of rings.
For ring homomorphisms of type ii), note that if $\varphi(\overline{1}) = k \in \mathbb{Z}$, then $n \varphi(\overline{1}) = 0$, which means $\varphi$ is the $0$ function. If you require that ring homomorphisms send 1 to 1, then there are no such ring homomorphisms.
For iii), we have that $\mathbb{Z} \to \mathbb{Z}_m$ must factor through $\mathbb{Z}_n \to \mathbb{Z}_m$ - which means that $(n) \subseteq (m)$ - i.e., $m$ divides $n$.