Proposition 3

Let M be an A-module and N a submodule of M. Then M is Artinian (resp. Noetherian) if and only if N and $M/N$ are.

We will give the proof in the case of Artinian modules; the case of Noetherian modules is analogous.

Suppose that M is Artinian. Since every submodule of N is a submodule of M, the module N is Artinian. Let $(P_n)_{n\in\mathbf{N}}$ be a decreasing sequence of submodules of $M/N$. There exists a decreasing sequence $(Q_n)_{n\in\mathbf{N}}$ of submodules of M containing N such that $P_n= Q_n/N$ for every $n\in \mathbf{N}$ (I, §4, No. 6, p. 41, Theorem 4). Since M is Artinian, the sequence $(Q_n)$ is stationary, hence so is the sequence $(P_n)$. Consequently, the module $M/N$ is Artinian.

Conversely, suppose that the modules N and $M/N$ are Artinian, and consider a decreasing sequence $(P_n)$ of submodules of M. The sequence $P'_n= N\cap P_n$ of submodules of N is stationary. Likewise, the sequence $P''_n= (N + P_n)/N$ of submodules of $M/N$ is stationary. Hence, there exists an integer $m\in \mathbf{N}$ such that we have $P'_n= P'_m$ and $P''_n= P''_m$ for every integer $n\geqslant m$. The sequence $(P_n)$ is then stationary by the following lemma.