Exercise 15, § 2

An $A$-module homomorphism $u: E \to F$ is called essential if it is injective and if, for every submodule $P \neq \{0\}$ of $F$, $u^{-1}(P) \neq \{0\}$; it suffices that this condition holds for every monogenous submodule $P \neq \{0\}$ of $F$. If $E$ is a submodule of $F$, $F$ is called 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, so is $v \circ u$. Conversely, if $v \circ u$ is essential and $v$ 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 $E$, so is $F$.

(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. (Prove it first 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 $\mathbf{Q}^{\mathbf{N}}$ is not an essential extension of $\mathbf{Z}^{\mathbf{N}}$.

(e) Let $E$ be a submodule of a module $F$. Show that there exists a submodule

Q of F such that Q ∩ E = {0} and that the restriction to E of the canonical homomorphism F → F/Q is essential.