Exercise 1, § 10

(a) Let $E$ be a right $A$-module. On the additive group $E^r$ (with $r \geq 1$) an external law of composition is defined with $\mathbf{M}_r(A)$ as set of operators, denoting by $x.P$, for every element $x = (x_i)_{1 \leq i \leq r}$ of $E^r$ and every square matrix $P = (\alpha_{ij}) \in \mathbf{M}_r(A)$, the element $y = (y_i)$ of $E^r$ such that

This external law defines with the additive law on $E^r$ a right $\mathbf{M}_r(A)$-module structure on $E^r$; by restricting the ring of operators of the ring of scalar matrices $I.\alpha$, the product $A$-module structure on $E^r$ is recovered. Show that for the $A$-module $E$ to have a generating system of $r$ elements, it is necessary and sufficient that the $\mathbf{M}_r(A)$-module $E^r$ be monogenous.

(b) Let $(x_i)_{1 \leq i \leq n}$ be a generating system of $E$ and $(y_j)_{1 \leq j \leq m}$ a family of elements of $E$ (resp. a generating system of $E$). Let $z, z', z''$ be three elements of $E^{m+n}$ such that $z_i = x_i$ for $1 \leq i \leq n$, $z_{n+j} = 0$ for $1 \leq j \leq m$, $z'_i = x_i$ for 1 $\leq i \leq n,\ z'_{n+j} = y_j$ for $1 \leq j \leq n,\ z''_i = 0$ for $1 \leq i \leq n,\ z''_{n+j} = y_j$ for $1 \leq j \leq n$. Show that there exist two invertible matrices $P, Q$ of $\mathbf{M}_{n+m}(\mathbf{A})$ such that $z' = z.P$ (resp. $z'' = z.Q$).

(c) If $\mathbf{A}$ is commutative, $u_P : x \mapsto x.P$ is an endomorphism of the $\mathbf{A}$-module $E'$ and the mapping $P \mapsto u_P$ is a homomorphism of the ring $\mathbf{M}_r(\mathbf{A})$ into the ring $\operatorname{End}_\mathbf{A}(E')$. If $E$ is a faithful $\mathbf{A}$-module this homomorphism is injective.