Answer to Question #107318 in Abstract Algebra for Sanjana

Question

Minimal Modules exist in Artinian Modules ?

Answer 1

Yes, it always exists.

If we assume that there is not a minimal submodule, then we can consruct a sequence of submodules $\{M_n\}_{n\in\mathbb N}$ such that $M_n\supset M_{n+1}$ and $M_n\neq M_{n+1}$ for every $n\in\mathbb N$. Indeed, let $M_1$ be arbitrary submodule.

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

Then our module is not artinian.