Finite modules over Artinian Rings are Artinian

Question

This question was left as an exercise in my class of Commutative algebra and I am struck on it.

Question: Prove that finite modules over artinian rings are artinian.

Thoughts: If ring is artinian then any descending chain of ideal stabilizes. A module is finitely generated means it has a finite generating set. Let I take an ascending chain of submodules: $M_1>...> M_n>...$. I am not able to understand how should I prove that this chain stabilizes. What relation the submodules have with the ideals? Module is an abelian group (M,+) with a binary operation $R\times M \to M$ satisfying some conditions and R is artinian so any descending chain stabilizes.

Can you please help me complete this proof?

Answer 1

Here is a sketch of a proof.

1. If $0 \to M \to N \to P \to 0$ and both $M,P$ are Artinian $R$-module, then $N$ is Artinian $R$-module.(Hint: if $R_i$ is a DC in $N$, consider its image in $P$, then consider its preimage in $M$.)

2. Since $R$ is Artinian $R$-module, $R^n$ is Artinian $R$-module by 1.

3. For any finitely generated module $M$, we have surjection $R^n \to M \to 0$, so $M$ is Artinian.