Exercise 16, § 2

A submodule E of an A-module F is said to be irreducible with respect to F (or in F) if $E \neq F$ and if E is not the intersection of two submodules of F distinct from E. a) For an A-module F to be an essential extension of each of its submodules $\neq \{0\}$, it is necessary and sufficient that $\{0\}$ be irreducible with respect to F. b) Let $(E_\lambda)_{\lambda \in L}$ be a family of submodules of an A-module F. We say that $(E_\lambda)_{\lambda \in L}$ is a reduced irreducible decomposition of $\{0\}$ in F if each of the $E_\lambda$ is irreducible in F, if the intersection of the $E_\lambda$ is reduced to 0 and if none of the $E_\lambda$ contains the intersection of the $E_\mu$ for $\mu \neq \lambda$. When this is the case, prove that the canonical homomorphism of F into $\bigoplus_{\lambda \in L} (F/E_\lambda)$ is essential (if $E'_\lambda = \bigcap_{\mu \neq \lambda} E_\mu$, observe that the image $F'_\lambda$ of $E'_\lambda$ in $F/E_\lambda$ is not zero, and that the image of F in $\bigoplus_{\lambda \in L} (F/E_\lambda)$ contains $\bigoplus_{\lambda \in L} F'_\lambda$; then apply exerc. 15 c)). Conversely, if $(E_\lambda)_{\lambda \in L}$ is a family of submodules of F irreducible in F and if the canonical homomorphism $F \to \bigoplus_{\lambda \in L} (F/E_\lambda)$ is essential, the family $(E_\lambda)$ is a reduced irreducible decomposition of $\{0\}$ in F.