# Given a split exact sequence $0 \to N \to M \to M \to 0$, when can we say $N=0$?

by user102248   Last Updated May 16, 2019 04:20 AM

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 ?) ?

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