# Noetherian and Artinian properties are inherited in short exact sequences

###### Theorem 1.

Let $M,M^{\prime},M^{\prime\prime}$ be $A$-modules and $0\to M^{\prime}\overset{\iota}{\to}M\overset{\pi}{\to}M^{\prime\prime}\to 0$ a short exact sequence. Then

1. 1.

$M$ is Noetherian if and only if $M^{\prime}$ and $M^{\prime\prime}$ are Noetherian;

2. 2.

$M$ is Artinian if and only if $M^{\prime}$ and $M^{\prime\prime}$ are Artinian.

For $\Leftarrow$, we will need a lemma that essentially says that a submodule of $M$ is uniquely determined by its image in $M^{\prime\prime}$ and its intersection with $M^{\prime}$:

###### Lemma 1.

In the situation of the theorem, if $N_{1},N_{2}\subset M$ are submodules with $N_{1}\subset N_{2}$, $\pi(N_{1})=\pi(N_{2})$, and $N_{1}\cap\iota(M^{\prime})=N_{2}\cap\iota(M^{\prime})$, then $N_{1}=N_{2}$.

###### Proof.

The proof is essentially a diagram chase. Choose $x\in N_{2}$. Then $\pi(x)=\pi(x^{\prime})$ for some $x^{\prime}\in N_{1}$, and thus $\pi(x-x^{\prime})=0$, so that $x-x^{\prime}\in\operatorname{im}\iota$, and $x-x^{\prime}\in N_{2}$ since $N_{1}\subset N_{2}$. Hence $x-x^{\prime}\in N_{2}\cap\iota(M^{\prime})=N_{1}\cap\iota(M^{\prime})\subset N% _{1}$. Since $x^{\prime}\in N_{1}$, it follows that $x\in N_{1}$ so that $N_{1}=N_{2}$. ∎

###### Proof.

($\Rightarrow$): If $M$ is Noetherian (Artinian), then any ascending (descending) chain of submodules of $M^{\prime}$ (or of $M^{\prime\prime}$) gives rise to a similar sequence in $M$, which must therefore terminate. So the original chain terminates as well.
($\Leftarrow$): Assume first that $M^{\prime},M^{\prime\prime}$ are Noetherian, and choose any ascending chain $M_{1}\subset M_{2}\subset\dots$ of submodules of $M$. Then the ascending chain $\pi(M_{1})\subset\pi(M_{2})\subset\dots$ and the ascending chain $M_{1}\cap\iota(M^{\prime})\subset M_{2}\cap\iota(M^{\prime})\subset\dots$ both stabilize since $M^{\prime}$ and $M^{\prime\prime}$ are Noetherian. We can choose $n$ large enough so that both chains stabilize at $n$. Then for $N\geq n$, we have (by the lemma) that $M_{N}=M_{n}$ since $\pi(M_{N})=\pi(M_{n})$ and $M_{N}\cap\iota(M^{\prime})=M_{n}\cap\iota(M^{\prime})$. Thus $M$ is Noetherian. For the case where $M$ is Artinian, an identical proof applies, replacing ascending chains by descending chains. ∎

## References

• 1 M.F. Atiyah, I.G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley 1969.
Title Noetherian and Artinian properties are inherited in short exact sequences NoetherianAndArtinianPropertiesAreInheritedInShortExactSequences 2013-03-22 19:11:52 2013-03-22 19:11:52 rm50 (10146) rm50 (10146) 5 rm50 (10146) Theorem msc 16D10