Answer to Question #107318 in Abstract Algebra for Sanjana

Question #107318
Minimal Modules exist in Artinian Modules ?
1
Expert's answer
2020-04-01T14:41:59-0400

Yes, it always exists.

If we assume that there is not a minimal submodule, then we can consruct a sequence of submodules {Mn}nN\{M_n\}_{n\in\mathbb N} such that MnMn+1M_n\supset M_{n+1} and MnMn+1M_n\neq M_{n+1} for every nNn\in\mathbb N. Indeed, let M1M_1 be arbitrary submodule.

Now suppose that MkM_k is constructed. Since it is not a minimal submodule, there is submodule NMkN\subset M_k. So take NN as Mk+1M_{k+1}. So Mk+1M_{k+1} is consructed.

Then our module is not artinian.


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