Exercise 16, § 2

A submodule E of an A-module F is called irreducible with respect to F (or in F) if E ≠ F and 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 ≠ {0} it is necessary and sufficient that {0} be irreducible with respect to F.

(b) Let (E_λ)_{λ ∈ L} be a family of submodules of an A-module F. (E_λ)_{λ ∈ L} is called a reduced irreducible decomposition of {0} in F if each of the F_λ is irreducible in F, if the intersection of the E_λ reduces to 0 and if none of the E_λ contains the intersection of the E_μ for μ ≠ λ. When this is so, show 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'_λ of E'_λ in F/E_λ is non-zero and that the image of F in $\bigoplus_{\lambda \in L} (F/E_\lambda)$ contains $\bigoplus_{\lambda \in L} F'_\lambda$; then apply Exercise 15(c)). Conversely, if (E_λ)_{λ ∈ L} is a family of submodules of F irreducible in F and the canonical homomorphism F → $\bigoplus_{\lambda \in L} (F/E_\lambda)$ is essential, the family (E_λ) is a reduced irreducible decomposition of {0} in F.