Exercise 14, § 3

Let $A$ be a ring, which is not necessarily commutative and whose Jacobson radical $m$ is such that $A/m$ is a field, and $M$ a finitely generated free $A$-module.

(a) Let $M'$ be another finitely generated free $A$-module and $u : M \to M'$ a homomorphism such that $u(M)$ is a direct factor of $M'$; the number $\dim(M)$ is called the \emph{rank} of $u$ and denoted by $\mathrm{rg}(u)$. For such a homomorphism $\mathrm{Ker}(u)$ is a direct factor of $M$ and $\mathrm{rg}(u) = \dim(M) - \dim(\mathrm{Ker}(u))$; for $u$ to be injective (resp. surjective), it is necessary and sufficient that $\mathrm{rg}(u) = \dim(M)$ (resp. $\mathrm{rg}(u) = \dim(M')$). The transpose ${}^t u$ is such that ${}^t u({M'}^*)$ is a direct factor of $M^*$ and $\mathrm{rg}({}^t u) = \mathrm{rg}(u)$.