What are the module homomorphisms from $\mathbb Q$ to $\mathbb Z/2\mathbb Z$?

Question

What are the module homomorphisms from $\mathbb Q$ to $\mathbb Z/2\mathbb Z$, looking at $\mathbb Q$ as a $\mathbb Z$-module? The impression I got from the proof in the book is that it is the zero homomorphism, but I do not understand why. Can someone explain this to me, please?

Answer 1

You're right, in that the only $\Bbb{Z}$-module morphism is the zero morphism. To see why, take any such morphism $f: \Bbb{Q} \to \Bbb{Z}/2\Bbb{Z}$ and a rational $q \in \Bbb{Q}$. Then we can write $q = 2q'$ for some other rational $q'$. Because $f$ is a $\Bbb{Z}$-module morphism, it is $\Bbb{Z}$-linear: $$ f(q) = f(2q') = 2f(q') = 0 $$ since $\Bbb{Z}/2\Bbb{Z}$ is a ring of characteristic 2.