Definition 1

The rational rank of a commutative group G is the dimension of the vector Q-space $G \otimes_{\mathbf{Z}} \mathbf{Q}$.

This dimension may also be defined as the least upper bound (finite or infinite) of the cardinals $r$ such that there exist $r$ elements of G which are linearly independent over $\mathbf{Z}$ (Algebra, Chapter II, § 7, no. 10, Proposition 26). The rational rank of G is zero if and only if G is a torsion group. For a subgroup of an additive group $\mathbf{R}^n$, the notion of rational rank coincides with that defined in General Topology, Chapter VII, § 1.

In the rest of this paragraph, we shall denote by $r(G)$ the rational rank of the commutative group G. If $G'$ is a subgroup of G, then (since $\mathbf{Q}$ is a flat $\mathbf{Z}$-module) we have the additive equation