Corollary 3

Let E be an A-module and g an A-linear mapping of E into a vector space F over K such that g(E) generates F and Ker(g) ⊂ T(E). Then the K-linear mapping $\bar{g}$ associated with g is an isomorphism of E_{(K)} onto F.

$\bar{g}$ is injective by Corollary 2 and the hypothesis that $g(E)$ generates F implies that $\bar{g}$ is surjective.

For every A-module E, the vector space E_{(K)} is said to be associated with E. For every subset S of E the rank of S over K (or by an abuse of language, the rank of S) is the rank of the canonical image $\phi(S)$ of S in E_{(K)}, in other words (no. 2, Definition 1) the dimension over K of the vector subspace of E_{(K)} generated by $\phi(S)$.

When E is a torsion-free A-module, it is usually identified with its canonical image $\phi(E)$ in E_{(K)}. With this convention, every generating system of E contains a basis of E_{(K)} (no. 1, Theorem 2). In particular: