I have to show that
$$\varphi:\mathbb{Z}/n\mathbb{Z}\rightarrow \mathbb{Z}/m\mathbb{Z},a+n\mathbb{Z}\mapsto a+m\mathbb{Z}$$
is a well-defined and surjective ring homomorphism for $m|n$.
My idea was to look at the map $\psi :\mathbb{Z}\rightarrow\mathbb{Z}/m\mathbb{Z},a\mapsto a+m\mathbb{Z}$ which is clearly surjective and then to use the fundemental theorem on homomorphisms.
Because $m|n$, the ideal $n\mathbb{Z}$ is a subset of kernel of $\psi$.
That tells me that $\varphi$ is surjective because of the above mentioned theorem.
But is this enough to say that $\varphi$ is well-defined or do I have to show that differently?
Best Answer
It suffices to show that $n\mathbb{Z}\subseteq \ker(\psi)$ because then $\psi$ factors through a unique (well-defined) homomorphism $\phi:\mathbb{Z}/n\mathbb{Z}\to \mathbb{Z}/m\mathbb{Z}$ given by $a+n\mathbb{Z}\mapsto a+m\mathbb{Z}$.
On the other hand, you can also check directly. Suppose that $a\equiv b \pmod{n}$. Then $n\mid (a-b)$ and hence $m\mid(a-b)$. So, $a\equiv b \pmod{m}$. So, $$\phi(a)=a\equiv b=\phi(b)\pmod{m}.$$ Hence, $\phi$ is well-defined.