Exercise 24, § 5

Let $A$ be a ring such that Spec(A) is connected.

a) If $M$ is an a-module projective of rank $n$, every direct factor $N$ of $M$ not reduced to 0 admits a rank $\leq n$.

b) Let $u$ be a homomorphism of a finite rank projective A-module $M$ into a finite rank projective A-module $M'$, such that $u(M)$ is a direct factor of $M'$. Now prove that Ker($u$) is then a direct factor of $M$ (cf. exerc. 23), and one has

Moreover the transpose $^t u$ is such that $^t u({M'}^*)$ is a direct factor of M* and $\operatorname{rg}(^t u({M'}^*)) = \operatorname{rg}(u(M))$ (cf. § 3, exerc. 14 a)).