My Question: A non-cyclic Artinian module has at least two distinct minimal (simple) submodule.
My attempt: Let $M$ be a non-cyclic Artinian $R$-module. Then there exists two nonzero element $x,y\in M$ such that $Rx\nsubseteq Ry$ and $Ry\nsubseteq Rx$. Hence, by Zorns Lemma, $M$ has two distinct minimal submodules as $M$ is Artinian.
What is wrong hare?
Firstly, you are taking a lot of liberties with your assertion.
Why do such $x,y$ exist?
Then why do you think that means there are distinct minimal submodules? It isn’t clear.
Why are you using Zorn’s lemma? You have both the DCC, already: you don’t need Zorn. And it’s not clear how you were applying it anyway.
—-
Let $R=\mathbb R[x,y,z]/(x^2,y^2,xz,yz,z^2-xy)$. It is known that $R$ is Artinian and has a simple socle, and hence a unique minimal ideal contained in all other nonzero ideals. It is also not a principal ideal ring. So if we take $M$ to be any nonprincipal ideal, we have a counterexample.
—- Eric Wofsey points out in my comments:
$\mathbb R[x,y]/(x^2,y^2)$ works for the same reasons!