An epimorphism from $\mathbb Z⊕\mathbb Z⊕\cdots$ to $\mathbb Q$

Question

I want an explicit example of an epimorphism from $\mathbb Z⊕\mathbb Z⊕\cdots$ to $\mathbb Q$. Thanks.

Answer 1

Try $$(x_1,x_2,\ldots)\mapsto \sum_{k\in\mathbb N}\frac 1kx_k$$ Note that the sum is in fact finite.

Answer 2

Hint: index the copies of $\mathbb Z$ with the primes and send $(0,0,\dots,0,1,0, \dots)$ with $1$ on the copy indexed by $p$ to $p\in\mathbb Q$.

(Note: this is for $(\mathbb Q_{>0}, \cdot)$. The same idea works more generally for countable abelian groups, the only work is finding a bunch of generators.)

Edit: Here's the general idea: let $A = \{a_1, a_2, \dots\}$ be your countable abelian group. Construct an epimorphism $\bigoplus \mathbb Z \to A$ by sending $(0,\dots,0,1,0,\dots)$ with a $1$ on position $n$ to $a_n$.

Answer 3

$$ \bigoplus_{n\in\Bbb N}\Bbb Z\simeq\bigoplus_{\text{$p$ prime}}\bigoplus_{t\in\Bbb N}\frac1{p^t}\Bbb Z\stackrel{\Sigma}\longrightarrow\Bbb Q. $$ Here $\Sigma$ denotes the addition map.

Obviously this map has a big kernel.