Exercise 15, § 2

One says that a homomorphism $u : E \to F$ of $A$-modules is essential if it is injective and if, for every submodule $P \neq \{0\}$ of $F$, one has $\overline{u^1}(P) \neq \{0\}$; it suffices that this condition be verified for every monogenic submodule $P \neq \{0\}$ of $F$. If $E$ is a submodule of $F$, one says that $F$ is an essential extension of $E$ if the canonical injection $E \to F$ is essential. a) Let $u : E \to F$, $v : F \to G$ be two $A$-homomorphisms of $A$-modules. If $u$ and $v$ are essential, the same is true of $v \circ u$. Conversely, if $v \circ u$ is essential and if $v$ is injective, then $u$ and $v$ are essential.

b) Let E be a submodule of an A-module F, and let $(F_\lambda)_{\lambda \in L}$ be a family of submodules of F containing E, and whose union is F. Show that if each of the $F_\lambda$ is an essential extension of E, the same is true of F. c) Let $(u_\lambda)_{\lambda \in L}$ be a family of essential homomorphisms $u_\lambda : E_\lambda \to F_\lambda$. Show that the homomorphism $\bigoplus_{\lambda \in L} u_\lambda : \bigoplus_{\lambda \in L} E_\lambda \to \bigoplus_{\lambda \in L} F_\lambda$ is essential. (First prove it when L has two elements, then use b).) d) The $\mathbf{Z}$-module $\mathbf{Q}$ is an essential extension of $\mathbf{Z}$, but the product $\mathbf{Q}^N$ is not an essential extension of $\mathbf{Z}^N$. e) Let E be a submodule of a module F. Show that there exists a submodule Q of F such that $Q \cap E = \{0\}$ and that the restriction to E of the canonical homomorphism $F \to F/Q$ is essential.