Is there any general formula for ring homomorphism from $\mathbb{Z}_{n} \rightarrow \mathbb{Z}_{p} \times \mathbb{Z}_{q}$ if gcd$(p, q) \neq 1$ ?
I got the link :What are all the homomorphisms between the rings $\mathbb{Z}_{18}$ and $\mathbb{Z}_{15}$?
But I am finding it difficult to show that gcd$(p, q) \neq 1$ ?
Any any hints/solution ?
Hint: A ring homomorphism must send $1$ to an idempotent.
Let $\phi: \mathbb{Z}_{n} \to \mathbb{Z}_{p} \times \mathbb{Z}_{q}$ be a ring homomorphism. Let $\phi([1]_n)=e=(a,b)$. Then the additivity of $\phi$ implies $\phi([x]_n)=xe$. In particular, $ne=0$. When this is satisfied, the map $\phi([x]_n)=xe$ is well defined. The multiplicativity of $\phi$ implies $e^2=e$. So, $\phi$ is determined by the choice of $e$, which must have additive order dividing $n$ and be idempotent.
If you require that a ring homomorphism preserve units, then $e=(1,1)$ and the additive requirement $ne=0$ is the only one. Otherwise, you need to consider the idempotents in $\mathbb{Z}_{p} \times \mathbb{Z}_{q}$.