Why $\mathbb{Q}$-vector space has no $\mathbb{Q}$-torsion?

Question

Why $\mathbb{Q}$-vector space has no $\mathbb{Q}$-torsion ?

What does the notion of $\mathbb{Q}$-torsion technically mean ?

Answer 1

In general, if $M$ is a module over a ring $R$, we say that an element $m \in M \setminus \{0\}$ is of torsion if there exists $r \in R$ such that $r \cdot m = 0$. For example, if we consider $\mathbb{Z}/2\mathbb{Z}$ as a $\mathbb{Z}$-module, then

$$ 2 \cdot \bar{1} = \bar{2} = \bar{0}. $$

If $M$ has some torsion element, we say that $M$ has $R$-torsion. This can never happen when $R$ is a field: suppose that we have a $\Bbbk$-vector space $V$ and $v \in V, \lambda \in \Bbbk$ such that $\lambda \cdot v = 0$. Multipying by $\lambda^{-1}$ we obtain

$$ v = 1 \cdot v = (\lambda^{-1}\lambda) \cdot v = \lambda^{-1}(\lambda \cdot v) = \lambda^{-1} \cdot 0 = 0. $$