Sequential direct limit of multiplication maps on the integers ($\underrightarrow{\lim }(\mathbb{Z}\stackrel{a_0}{\to }\mathbb{Z}\stackrel{a_1}{\to }\cdots )$)

For nonzero integers $a_i$ and $P_n=a_0\cdots a_{n-1}$, the compatible embeddings of stage $n$ into $\mathbb{Q}$ send $m$ to $m/P_n$. They identify the direct limit with

The additive group ${\mathbb{Q}}_{\mathrm{sq}}$ consists of the rational numbers whose reduced denominators are square-free integers. If $p_0,p_1,\dots$ lists the primes without repetition, then