Proposition 3

A Noetherian or Artinian A-module M is the direct sum of a finite family of indecomposable submodules.

Let us first prove that every nonzero submodule P of M has an indecomposable direct factor. If this were not the case, then every direct factor submodule of P would be decomposable; proceeding by induction, we could then construct, for every $n\in \mathbf{N}$, nonzero submodules $N'_n$ and $N''_n$ of P such that $P = N'_0\oplus N''_0$ and $N'_{n-1}= N'_n\oplus N''_n$ for $n\geqslant 1$. But then, the sequence of submodules $N''_0+\cdots + N''_n$ would be strictly increasing, and that of the submodules $N'_n$ would be strictly decreasing. The module M would be neither Noetherian nor Artinian, contrary to the assumption.

Now, suppose that M is not the direct sum of a finite family of indecomposable submodules. By induction, we construct indecomposable submodules $P''_n$ of M and nonzero submodules $P'_n$ of M for every $n\in \mathbf{N}$ such that $M = P'_0\oplus P''_0$ and $P'_{n-1}= P'_n\oplus P''_n$ for $n\geqslant 1$. Indeed, M is nonzero, and the first part of the proof, applied to P = M, provides $P'_0$ and $P''_0$. Since the modules $P'_k$ and $P''_k$ are defined for $k < n$, by the first part of the proof, there exist submodules $P'_n$ and $P''_n$ such that $P'_{n-1}= P'_n\oplus P''_n$ with $P''_n$ indecomposable. The relation $M = P'_n\oplus P''_0\oplus \cdots \oplus P''_n$ then implies that $P'_n\not= 0$ because M is not the direct sum of a finite family of indecomposable modules.

The sequence of submodules $P''_0\oplus \cdots \oplus P''_n$ is strictly increasing, and the sequence of submodules $P'_n$ is strictly decreasing. This contradicts the assumption that M is Noetherian or Artinian.

The question of the uniqueness of the decomposition of a module as a direct sum of indecomposable submodules will be studied in the next subsection.