# $N$ submodule of $M$ and $N \cong M$ does not necessarily imply that $M=N$

Let $M, N$ be $A$-modules with $A$ being a commutative ring. Suppose that $N$ is a submodule of $M$ and also that $N$ is isomorphic to $M$. According to my understanding this does not necessarily imply that $M=N$. Is this statement accurate? If yes, at what kind of cases do we have this phenomenon?

-

Take $A = \mathbb{Z}$, $M = \mathbb{Z}$ and $N = 2 \mathbb{Z}$.

-

Here's an example: let $A$ be any ring, let $M=\bigoplus_{n=1}^\infty B$ for a non-trivial $A$-module $B$, and let $N$ be $$N=\{(a_i)\in M\mid a_1=0\}.$$ Then $N\neq M$, but the map $f:N\to M$ defined by $f((a_i))=(a_{i-1})$ is an isomorphism.

-

A module is called cohopfian if every injective endomorphism is surjective. A cohopfian module $M$ will not have any proper submodules isomorphic to $M$.
$M$ will be cohopfian if it is any of the following: