Proposition 17

For every A-module E and every mapping $f : T \to E$, there exists one and only one A-linear mapping $g : A_s^{(T)} \to E$ such that $f = g \circ \phi$.

The condition $f = g \circ \phi$ means that $g(e_t) = f(t)$ for all $t \in T$, which is equivalent to $g(\xi e_t) = \xi f(t)$ for all $\xi \in A$ and all $t \in T$ and also means that $g \circ j_t$ is the linear mapping $\xi \mapsto \xi f(t)$ of $A_s$ into $E$ for all $t \in T$; the proposition is therefore a special case of no. 6, Proposition 6.

The linear mapping $g$ is said to be determined by the family $(f(t))_{t \in T}$ of elements of $E$; by definition

The kernel $R$ of $g$ is the set of $(\xi_t) \in A_s^{(T)}$ such that $\sum_t \xi_t f(t) = 0$; it is sometimes said that the module $R$ is the module of linear relations between the elements of the family $(f(t))_{t \in T}$. The exact sequence

is said to be determined by the family $(f(t))_{t \in T}$.