Exercise 24, § 5

Let $A$ be a ring such that $\mathrm{Spec}(A)$ is connected.

(a) If $M$ is a projective $A$-module 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 projective $A$-module of finite rank $M$ to a projective $A$-module of finite rank $M'$ such that $u(M)$ is a direct factor of $M'$. Show that $\mathrm{Ker}(u)$ is then a direct factor of $M$ (cf. Exercise 23) and

Moreover the transpose ${}^t u$ is such that ${}^t u({M'}^*)$ is a direct factor of $M^*$ and $\mathrm{rg}({}^t u({M'}^*)) = \mathrm{rg}(u(M))$ (cf. § 3, Exercise I4 (a)).