202507061507

Tags : Ring Theory, Module Theory

Ascending Chain Condition

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Theorem

Let $R$ be a commutative ring, and let $M$ be an $R$-module, then the following are equivalent.

• $M$ is Noetherian
• Every ascending chain of submodules of $M$ stabilizes, this is called the Ascending Chain Condition
• Every nonemtpy family of sub-modules of $M$ hsa a maximal element wrt inclusion.

2 $\Rightarrow$ 3 is trivial. for 1 $\Rightarrow$ 2, consider the following ascending chain of sub-modules of $M$:

$$N_1\subseteq N_2\subseteq N_3\subseteq \dots$$

Let $N={\bigcup }_i{N}_i$ and set $N=⟨n_1,n_2\dots n_k⟩$. We know that each $n_i\in N_j$ for some $j$. Pick the largest such $j$ so we have $N_j$ contains all generators of $N$, thus $N_j=N$.

For $3\Rightarrow 1$, let $N$ be a submodules of $M$, then the family of finitely generated submodules of $N$ is non-empty and hence has a maximal element $N^{\prime }$. If $N^{\prime }$ is not equal to $N$ then we can add another element to it and we can get a bigger finitely generated sub-modules of $N$. Thus $N$ is finitely generated.

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References

• Noetherian Modules
• Noetherian Ring and PIDs