Exercise 17, § 2

a) Let M, N, M’, N’ be four submodules of a module E, such that $M \cap N = M' \cap N' = P$. Now prove that one has

Deduce that if M is irreducible in E, one necessarily has $P = M' \cap N$ or $P = N' \cap N$. b) Let $(N_i)_{1 \leq i \leq n}$ be a finite family of submodules of E irreducible in E, and let M be its intersection. Now prove that if $(P_j)_{1 \leq j \leq m}$ is a finite family of submodules of E such that $M = \bigcap_j P_j$, then, for every index i such that $1 \leq i \leq n$, there exists an index $\varphi(i)$ such that $1 \leq \varphi(i) \leq m$ and such that, if one sets $N'_i = \bigcap_{k \neq i} N_k$, one has $M = P_{\varphi(i)} \cap N'_i$ (use a)). Deduce that if none of the $P_j$ contains the intersection of the $P_k$ of index $\neq j$, one has $m \leq n$ (replace successively the $N_i$ by suitable $P_j$, using the preceding result). Conclude that two irreducible decompositions of M in E, one of which is finite, necessarily have the same number of terms (cf. II, p. 188, Exercise 23).