How to find all ring homomorphisms from $\mathbb Z_{12} \to \mathbb Z_{30}$?
I know that it is enough to determine $f([1]_{12})$ ; moreover $f([1]_{12})$ should be an idempotent element of $\mathbb Z_{30}$ and $\text{ord}(f([1]_{12})\mid 12$ and $30$, so we must find the idempotents in $\mathbb Z_{30}$ whose additive orders divide $\gcd(12,30)=6$. But this still leaves a lot to check. Is there any further restrictions on homomorphisms of such type?
I don't know why the OP has thought that there is "a lot to check". There are only six possibilities: $f(1)\in\{0,5,10,15,20,25\}$ and among these the idempotents are $\{0,10,15,25\}$, so there are four ring homomorphisms.