example of an Artinian module which is not Noetherian
Of course each is a submodule and it is easy to see, that
Let be the canonical projection. Then is a submodule of and
The inclusions are proper, because for any we have
due to Third Isomorphism Theorem for modules. This shows, that is not Noetherian.
In order to show that is Artinian, we will show, that each proper submodule of is of the form . Let be a proper submodule. Assume that for some and we have
We may assume that . Therefore there are such that
Now, since is a -module we have
and since we have that
Now, let be the smallest number, such that . What we showed is that
because for every (and only for such ) we have and thus is a image of a submodule of , which is generated by and this is precisely . Now let
This chain has to stabilize, which completes the proof.
|Title||example of an Artinian module which is not Noetherian|
|Date of creation||2013-03-22 19:04:18|
|Last modified on||2013-03-22 19:04:18|
|Last modified by||joking (16130)|