Missing a detail about Chinese Remainder Theorem and $Z$ Ring isomorphisms.

Question

I'm trying to prove that $\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/n\mathbb{Z} \cong \mathbb{Z}/mn\mathbb{Z}$ holds only when $\gcd(m,n)=1$ or in simpler terms when $n,m$ are coprime integers. So far I understand how a simple construction can be used when the condition is met to show that there exists a bijection such as: $$\psi\colon \mathbb{Z}/mn\mathbb{Z}\rightarrow \mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}/m\mathbb{Z} \ \psi(x)=(x \bmod n, x \bmod m)$$ However what I'm missing here is where exactly do we need the requirement that $\gcd(m,n)=1$?

Answer 1

Firstly observe that the kernel of the map defined from $\mathbb{Z}$ is $m\mathbb{Z}\cap n\mathbb{Z}$ which is equal to $mn\mathbb{Z}$ only when $m,n$ are coprime.

Secondly try to show that the map is surjective. Here also you will require the comaximality condition.