Let $M$ be a module over a commutative ring $R$. Let $N$ be a submodule of $M$ such that there is a split exact sequence $0 \to N \to M \to M \to 0$ . So in particular ,$M \cong M \oplus N$.
Under what additional conditions on $M,N$ or $R$, can we say that $N=0$ ?
Of course if $M$ is finitely generated, then $N=0$. What other conditions on $M$ or $N$ or $R$ would make that true (may be assuming $R$ Noetherian and something more ?) ?