Find all homomorphisms from a semigroup $(\mathbb{Z},+) \to (\mathbb{Q},+)$

Question

In the book Karpfinger & Meyberg (2013) "Algebra" I encountered this problem. So far I have figured out $\tau(x) = qx$ wehre $q \in \mathbb{Q}$. Are there other homomorphisms? Is there a general method to ideitify such homomorphisms?

Answer 1

Note that any homomorphism $f:\mathbb{Z}\rightarrow \mathbb{Q}$ is determined by where $1$ is sent. Suppose $1\mapsto q\in\mathbb{Q}$, then our homomorphism is defined by $f(n) = nf(1) = nq$. Therefore we found the homomorphisms you had already found and proved that these are the only ones.

Answer 2

Let $f: \mathbb{Z} \to \mathbb{Q}$ be a semigroup homomorphism. Let $a = f(0)$. Since $0 = 0 + 0$, we get $a = f(0) = f(0 + 0) = f(0)+f(0) = a + a\ $, whence $a = 0$.

Next, since $n + (-n) = 0$, we get $0 = f(0) = f(n + (-n)) = f(n) + f(-n)$, whence $f(-n) = -f(n)$ for all $n$.

You can now use Marc's answer to conclude.