Noetherian and Artinian properties are inherited in short exact sequences

Theorem 1.

Let M,M,M′′ be A-modules and 0M𝜄M𝜋M′′0 a short exact sequenceMathworldPlanetmathPlanetmath. Then

  1. 1.

    M is NoetherianPlanetmathPlanetmathPlanetmath if and only if M and M′′ are Noetherian;

  2. 2.

    M is Artinian if and only if M and M′′ are Artinian.

For , we will need a lemma that essentially says that a submoduleMathworldPlanetmath of M is uniquely determined by its image in M′′ and its intersectionDlmfPlanetmath with M:

Lemma 1.

In the situation of the theorem, if N1,N2M are submodules with N1N2, π(N1)=π(N2), and N1ι(M)=N2ι(M), then N1=N2.


The proof is essentially a diagram chase. Choose xN2. Then π(x)=π(x) for some xN1, and thus π(x-x)=0, so that x-ximι, and x-xN2 since N1N2. Hence x-xN2ι(M)=N1ι(M)N1. Since xN1, it follows that xN1 so that N1=N2. ∎


(): If M is Noetherian (Artinian), then any ascending (descending) chain of submodules of M (or of M′′) gives rise to a similar sequence in M, which must therefore terminate. So the original chain terminates as well.
(): Assume first that M,M′′ are Noetherian, and choose any ascending chain M1M2 of submodules of M. Then the ascending chain π(M1)π(M2) and the ascending chain M1ι(M)M2ι(M) both stabilize since M and M′′ are Noetherian. We can choose n large enough so that both chains stabilize at n. Then for Nn, we have (by the lemma) that MN=Mn since π(MN)=π(Mn) and MNι(M)=Mnι(M). Thus M is Noetherian. For the case where M is Artinian, an identical proof applies, replacing ascending chains by descending chains. ∎


  • 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
Canonical name NoetherianAndArtinianPropertiesAreInheritedInShortExactSequences
Date of creation 2013-03-22 19:11:52
Last modified on 2013-03-22 19:11:52
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 5
Author rm50 (10146)
Entry type Theorem
Classification msc 16D10