Definition 10

A family $(a_t)_{t \in T}$ of elements of an $\mathbf{A}$*-module* $E$ is called a free family (resp. a basis of $E$) if the linear mapping $A_s^{(T)} \to E$ determined by this family is injective (resp. bijective). A module is called free if it has a basis.

In particular, a commutative group $G$ is called free if $G$ (written additively) is a free $\mathbf{Z}$*-module* (cf. I, § 7, no. 7).

Definition 10, together with Corollary 2 to Proposition 17, show that a basis of an $\mathbf{A}$-module $E$ is a free generating family of $E$. Every free family of elements of $E$ is thus a basis of the submodule it generates.

By definition, the $\mathbf{A}$-module $A_s^{(T)}$ is free and the family $(e_t)_{t \in T}$ is a basis (called canonical) of this $\mathbf{A}$-module. When $\mathbf{A} \neq \{0\}$, $T$ is often identified with the set of $e_t$ by the canonical bijection $t \mapsto e_t$; this amounts to writing $\sum_{t \in T} \xi_t \cdot t$ instead of $\sum_{t \in T} \xi_t a_t$ for the elements of $A_s^{(T)}$. When this convention is adopted, the elements of $A_s^{(T)}$ are called formal linear combinations (with coefficients in $\mathbf{A}$) of the elements of $T$.

Definition 10 and Proposition 17 give immediately the following result: