Corollary 3

Let $E$ be a free $\mathbf{A}$*-module*, $(a_t)_{t \in T}$ a basis of $E$, $F$ an $\mathbf{A}$*-module and* $(b_t)_{t \in T}$ a family of elements of $F$. There exists one and only one linear mapping $f : E \to F$ such that

For $f$ to be injective (resp. surjective), it is necessary and sufficient that $(b_t)$ be a free family in $F$ (resp. a generating system of $F$).

When a family $(a_t)_{t \in T}$ is not free, it is called related. Definition 10 may also be expressed as follows: to say that the family $(a_t)_{t \in T}$ is free means that the relation $\sum_{t \in T} \lambda_t a_t = 0$ (where the family $(\lambda_t)$ is of finite support) implies $\lambda_t = 0$ for all $t \in T$; to say that $(a_t)_{t \in T}$ is a basis of $E$ means that every $x \in E$ can be written in one and only one way in the form $x = \sum_{t \in T} \xi_t a_t$; for all $t \in T$, $\xi_t$ is then called the component (or coordinate) of $x$ of index $t$ with respect to the basis $(a_t)$; the mapping $x \to \xi_t$ from $E$ to $A_s$ is linear.

Suppose $\mathbf{A} \neq \{0\}$; then, in an $\mathbf{A}$-module $E$, two elements of a free family $(a_t)_{t \in T}$ whose indices are distinct are themselves distinct: for if $a_{t'} = a_{t''}$ for $t' \neq t''$, then $\sum_{t \in T} \lambda_t a_t = 0$ with $\lambda_{t'} = 1$, $\lambda_{t''} = -1$ and $\lambda_t = 0$ for the elements of $T$ distinct from $t'$ and $t''$. A subset $S$ of $E$ will be called a free subset (resp. a basis of $E$) if the family defined by the identity mapping of $S$ onto itself is free (resp. a basis of $E$); every family defined by a bijective mapping of an indexing set onto $S$ is then free (resp. a basis). The elements of a free subset of $E$ are also called linearly independent.

If a subset of E is not free, it is called related or a related system and its elements are said to be linearly dependent.

Every subset of a free subset is free; in particular, the empty subset is free and is a basis of the submodule {0} of E.