If $M/N\cong M$, can we conclude that $N=0$?

Question

Let $M$ be an $R$-module, where we may assume that $R$ is an integral domain.

Let $N$ be a submodule of $M$.

Suppose that $M/N\cong M$. Can we conclude that $N=0$?

(If no, what are some sufficient conditions that make it true?)

Update: I learn that for "infinite dimensional" cases it can fail. How about when $M$ is finitely generated, does it work?

Thanks a lot.

Answer 1

It fails for vector spaces of infinite dimension.

Answer 2

It is not true. Let $M = \bigoplus_{n=1}^\infty \mathbb{Z}$, and let $N = \{(z,0,0,0,\ldots)\,|\, z \in \mathbb{Z}\}$. Then $M/N \cong M$.