Why is $M/(N\cap P)$ Artinian when $M/N$ and $M/P$ are Artinian?

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In summary, an Artinian module is one that satisfies the descending chain condition. This relates to the question about $M/(N\cap P)$ as the quotient module must also be Artinian if both $M/N$ and $M/P$ are Artinian. The Artinian property is preserved under quotients because the submodules of the quotient correspond to submodules of the original module. An example of this is seen with $M/(N\cap P)$ where $M/N$ and $M/P$ are isomorphic to $\mathbb{Z}/3\mathbb{Z}$ and $M/(N\cap P)$ is isomorphic to $\mathbb{Z}/2\mathbb{
  • #1
Euge
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Here is this week's POTW:

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Let $R$ be a commutative ring. If $N$ and $P$ are submodules of an $R$-module $M$ such that $M/N$ and $M/P$ are Artinian, show that $M/(N\cap P)$ is Artinian.

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No one answered this week's problem. You can read my solution below.
$M/(N\cap P)$ is isomorphic to a submodule of the Artinian module $M/N \times M/P$ via the $R$-mapping $M/(N\cap P) \to M/N \times M/P$ given by $m + N\cap P \mapsto (m + N, m + P)$; hence, $M/(N\cap P)$ is Artinian.
 

1. What is the definition of an Artinian module?

An Artinian module is a module that satisfies the descending chain condition, meaning that any decreasing chain of submodules eventually stabilizes.

2. How does the definition of Artinian modules relate to the question about $M/(N\cap P)$?

The question is asking about the Artinian property of the quotient module $M/(N\cap P)$. In order for this quotient to be Artinian, both $M/N$ and $M/P$ must also be Artinian.

3. Why is the Artinian property preserved under quotients?

The Artinian property is preserved under quotients because if a module is Artinian, then any decreasing chain of submodules of the quotient will also eventually stabilize, since the submodules of the quotient correspond to submodules of the original module.

4. Can you provide an example to illustrate why $M/(N\cap P)$ is Artinian when $M/N$ and $M/P$ are Artinian?

Consider the module $M=\mathbb{Z}/6\mathbb{Z}$, and let $N=2\mathbb{Z}/6\mathbb{Z}$ and $P=3\mathbb{Z}/6\mathbb{Z}$. Both $M/N$ and $M/P$ are isomorphic to $\mathbb{Z}/3\mathbb{Z}$, which is Artinian. Then, $M/(N\cap P)$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$, which is also Artinian. This example shows that the Artinian property is preserved under quotients.

5. How does the Artinian property of $M/(N\cap P)$ relate to the submodules $N$ and $P$?

The Artinian property of $M/(N\cap P)$ implies that $N\cap P$ is a finite intersection of Artinian submodules of $M$. This means that $N$ and $P$ must also be Artinian, since any finite intersection of Artinian submodules is itself Artinian.

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