Artinian if and only if Noetherian

Let $R$ be a ring (commutative, with identity), $m$ a maximal ideal and $M$ an $R$-module. Suppose $m^nM=0$ for some $n>0$. Then

$M$ is Noetherian if and only if $M$ is Artinian

Do you have any idea how to solve this?

Solution 1:

We will use the following standard commutative algebra fact:

Let $A$ be a commutative ring, and $E$ be an $A$-module. Suppose there is a filtration $$E = E_0\supseteq E_1\supseteq\cdots\supseteq E_n = 0$$ of $A$-submodules $E_i$. Then $E$ is Noetherian (resp. Artinian) $\iff$ each quotient $E_i/E_{i+1}$ is Noetherian (resp. Artinian).

Note that you have a filtration $$M \supseteq mM\supseteq m^2M\supseteq\cdots\supseteq m^nM = 0.$$ Thus $M$ is Noetherian (resp. Artinian) if and only if each of the successive quotients $m^kM/m^{k+1}M$ are Noetherian (resp. Artinian) $R$-modules. Note that each quotient $m^kM/m^{k+1}M$ is naturally an $R/m$-module, and moreover that $m^kM/m^{k+1}M$ will be a Noetherian (resp Artinian) $R$-module $\iff$ it is a Noetherian (resp Artinian) $R/m$-module. Since $R/m$ is a field, we know that $m^kM/m^{k+1}M$ is a Noetherian $R/m$-module $\iff$ it is an Artinian $R/m$-module $\iff$ it is finite dimensional over $R/m$. In particular, each of the quotients $m^kM/m^{k+1}M$ is Noetherian if and only if it is Artinian, and hence $M$ is Noetherian if and only if it is Artinian.