$N$ 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?

To answer the half about "When can we expect this?":
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:


*

*finite

*Artinian

*Noetherian and injective

