Suppose $S$ is a submodule of the $R$-module $M$, then it is not necessarily possible to decompose $M$ as
$$ M = S \oplus S^c . $$
However, if we suppose that $M$ is finitely generated and Noetherian, and $S$ is a submodule, can we always find a submodule $S'$ that has a trivial intersection with $S$, such that $S \oplus S^c$ is a finite index subgroup of $M$ (When we view them as abelian groups)?
Because $M$ is Noetherian, I know we should be able to find a maximal submodule $S'$ such that $S$ and $S'$ have a trivial intersection. I am struggling to see if this implies that the direct sum has a finite index in $M$. Any hint would be really appreciated.
There are algebras over a field having infinite dimensional simple modules. For example, the first Weyl algebra $k\langle x,y\rangle/(yx-xy-1)$ in characteristic zero, having the simple module $k[x]$ where $y$ acts as differentiation. Taking a non-split extension $M$ of two such simples, it has length two and is indecomposable, so is noetherian with simple socle, which has infinite codimension.