example of an Artinian module which is not Noetherian

It is well known, that left (right) Artinian ring is left (right) NoetherianPlanetmathPlanetmathPlanetmath (Akizuki-Hopkins-Levitzki theorem). We will show that this no longer holds for modules.

Let be the ring of integersMathworldPlanetmath and the field of rationals. Let p be a prime numberMathworldPlanetmath and consider


Of course G is a -module via standard multiplicationPlanetmathPlanetmath and addition. For n0 consider


Of course each GnG is a submoduleMathworldPlanetmath and it is easy to see, that


where each inclusion is proper. We will show that G/ is ArtinianPlanetmathPlanetmath, but it is not Noetherian.

Let π:GG/ be the canonical projection. Then Gn=π(Gn) is a submodule of G/ and


The inclusions are proper, because for any n>0 we have


due to Third Isomorphism Theorem for modules. This shows, that G/ is not Noetherian.

In order to show that G/ is Artinian, we will show, that each proper submodule of G/ is of the form Gn. Let NG/ be a proper submodule. Assume that for some a and n0 we have


We may assume that gcd(a,pn)=1. Therefore there are α,β such that


Now, since N is a -module we have


and since 0+=β+=βpnpn+N we have that


Now, let m>0 be the smallest number, such that 1pm+N. What we showed is that


because for every 0nm-1 (and only for such n) we have 1pn+N and thus N is a image of a submodule of G, which is generated by 1pn and this is precisely Gm-1. Now let


be a chain of submodules in G/. Then there are natural numbersMathworldPlanetmath n1,n2, such that Ni=Gni. Note that GkGs if and only if ks. In particular we obtain a sequenceMathworldPlanetmathPlanetmath of natural numbers


This chain has to stabilize, which completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof.

Title example of an Artinian module which is not Noetherian
Canonical name ExampleOfAnArtinianModuleWhichIsNotNoetherian
Date of creation 2013-03-22 19:04:18
Last modified on 2013-03-22 19:04:18
Owner joking (16130)
Last modified by joking (16130)
Numerical id 5
Author joking (16130)
Entry type Example
Classification msc 16D10