Why can’t Zorn’s lemma be used to show any module has maximal submodule

Question

So you can use Zorn’s lemma to show that if $R$ is a ring then $R $ has a maximal left ideal. Show you would let $X$ be the set of all proper left ideals of $R$. Then can show any chain has an upper bound and so Zorn’s lemma shows that $X$ has a maximal element which is the maximal left ideal. I assume this proof is absolutely standard, but why can’t we just replace ideals with left submodules and instead of talking about ideals of $R$ we talk about submodules of $M$, say. What fails in the proof?