Find the linearly independent subsets of the rational additive group over $\displaystyle \mathbb{Z}$
Thanks!
$\displaystyle A \subset \mathbb{Q}$ is linearly independent over integers if and only if $\displaystyle A=\{r\},$ for some $\displaystyle 0 \neq r \in \mathbb{Q}.$ such sets are obviously linearly independent. now suppose $\displaystyle B \subset \mathbb{Q}$ is linearly independent over integers and
$\displaystyle |B| \geq 2.$ obviously $\displaystyle 0 \notin B.$ choose any two elements of $\displaystyle B$, say $\displaystyle r_1=\frac{a}{b}, \ r_2=\frac{c}{d}.$ then $\displaystyle bcr_1 - adr_2 =0,$ which proves that $\displaystyle \{r_1,r_2 \}$ and therefore $\displaystyle B$ is not linearly independent over integers. Q.E.D.