Given $f : \mathbb {Z}_{4} \rightarrow \mathbb{Z}_{10} $. find the number of ring homomorphism ?
My attempt : I got $4$ , my thinking is that the idempotents element in $\mathbb{Z}_{10}$ are $ \{ 0,1,5,6\}$, so there are $4$ ring homomorphism
Is its true ?
Since $\mathbb{Z}_4$ is cyclic, everything is determined by $f(1)$. Furthermore, you must have ord$(f(1)) \mid \mathbb{Z}_{10}$ by Lagrange's Theorem, so ord$(f(1)) \in \{1,2,5,10\}$. Now obviously it can't be $5$ or $10$. (Why?) Now show the other two work, giving 2 homomorphisms.