rational rank of a group

In the following, $G$ is an abelian group.

Definition 1.

The group $\mathrm{Q}{\mathrm{\otimes }}_{\mathrm{Z}}G$ is called the divisible hull of $G$.

It is a $\mathbb{Q}$-vector space such that the scalar $\mathbb{Z}$-multiplication of $G$ is extended to $\mathbb{Q}$.

Definition 2.

The elements $g_{\mathrm{1}}\mathrm{,}{g}_{\mathrm{2}}\mathrm{,}\mathrm{\dots }\mathrm{,}{g}_r\mathrm{\in }G$ are called rationally independent if they are linearly independent over $\mathrm{Z}$, i.e. for all $n_{\mathrm{1}}\mathrm{,}\mathrm{\dots }\mathrm{,}{n}_r\mathrm{\in }Z$:

$$n_1{g}_1+\mathrm{\dots }+n_r{g}_r=0\Rightarrow n_1=\mathrm{\dots }=n_r=0.$$

Definition 3.

The dimension of $\mathrm{Q}{\mathrm{\otimes }}_{\mathrm{Z}}G$ over $\mathrm{Q}$ is called the rational rank of $G$.

We denote the rational rank of $G$ by $r(G)$.
Example:
$r(\mathbb{Z}\times \mathbb{Z})=2$ because $\mathbb{Q}{\otimes }_{\mathbb{Z}}(\mathbb{Z}\times \mathbb{Z})=(\mathbb{Q}{\otimes }_{\mathbb{Z}}\mathbb{Z})\times (\mathbb{Q}{\otimes }_{\mathbb{Z}}\mathbb{Z})=\mathbb{Q}\times \mathbb{Q}$.
Properties:

• If $H$ is a subgroup of $G$ then we have:

  $$r(G)=r(H)+r(G/H).$$

  It results from the fact that ${}_{\mathbb{Z}}\mathbb{Q}$ is a flat module.

• The rational rank of the group $G$ can be defined as the least upper bound (finite or infinite) of the cardinals $r$ such that there exist $r$ rationally independent elements in $G$.