Proposition 20

Every A-module E is isomorphic to a quotient module of a free A-module.

If T is a generating set of E, there exists a surjective linear mapping $A_s^{(T)} \to E$ (Corollary 2 to Proposition 17), and if R is the kernel of this mapping, E is isomorphic to $A_s^{(T)}/R$.

In particular we may take $T = E$; then there is a surjective linear mapping $A_s^{(E)} \to E$, called canonical.

In particular, to say that an A-module E is finitely generated (no. 7) means that it is isomorphic to a quotient of a free A-module with a finite basis or also that there exists an exact sequence of the form

Note that if $A \neq \{0\}$ every basis of a finitely generated free module E is necessarily finite, for if S is a finite generating system and B a basis of E, each element of S is a linear combination of a finite number of elements of B and if B' is the set of all the elements of B which figure thus in the expression for the elements of S, B' is finite and every $x \in E$ is a linear combination of elements of B', hence $B' = B$.