Corollary

Under the hypotheses of Proposition 6 the following properties are equivalent:

(a) $u$ is an isomorphism of $M$ onto $N$; (b) $M$ and $N$ have the same rank (Algebra, Chapter II, § 7, no. 2) and $u$ is surjective; (c) $1 \otimes u: M/mM \to N/mN$ is bijective.

Clearly (a) implies (b); (b) implies that $1 \otimes u$ is surjective; moreover the hypothesis that M and N have the same rank implies that so do the vector spaces $(A/m) \otimes_A M$ and $(A/m) \otimes_A N$ over $A/m$, hence $1 \otimes u$ is bijective (Algebra, Chapter II, , no. 4, Corollary to Proposition 9) and (b) implies (c). Finally, condition (c) implies, by Proposition 6, that N is the direct sum of $u(M)$ and a free submodule P and $u$ is an isomorphism of M onto $u(M)$; if $P \neq 0$, then $(A/m) \otimes_A P \# 0$ and $1 \otimes u$ would not be surjective; hence (c) implies (a)

The propositions proved above in this no. will usually be applied when $A$ is a local ring and $m$ its maximal ideal. Corollary 2 to Proposition 5 is then completed by