Noetherian and Artinian properties are inherited in short exact sequences
In the situation of the theorem, if are submodules with , , and , then .
The proof is essentially a diagram chase. Choose . Then for some , and thus , so that , and since . Hence . Since , it follows that so that . ∎
(): If is Noetherian (Artinian), then any ascending (descending) chain of submodules of (or of ) gives rise to a similar sequence in , which must therefore terminate. So the original chain terminates as well.
(): Assume first that are Noetherian, and choose any ascending chain of submodules of . Then the ascending chain and the ascending chain both stabilize since and are Noetherian. We can choose large enough so that both chains stabilize at . Then for , we have (by the lemma) that since and . Thus is Noetherian. For the case where 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|
|Date of creation||2013-03-22 19:11:52|
|Last modified on||2013-03-22 19:11:52|
|Last modified by||rm50 (10146)|