I was wondering whether the following is true or not:
Suppose $N$ and $L$ are submodules of $M$ with $N\subseteq L$. If $M/L\cong M/N$ then $L=N$.
I am particularly curious about the case when $L=\ker \varphi$ for some $\varphi\in $Hom$(M,P)$ but the general case was also interesting to me.
My Attempt: I know that the isomorphism theorems give us that $$\frac{M/N}{L/N}\cong M/L\cong M/N,$$ but I wasn't sure if that was enough to say that $L/N=0$.
Thanks for any help.
If the isomorphism $M/L\cong M/N$ is canonical it is easy to show that $M=N$. But in general is not true see the example: $M=\Bbb R[X]$ and $N=\Bbb R_n[X]$ and $L=\Bbb R_{n+1}[X]$ (as $\Bbb R$ module $=$ $\Bbb R$ vector space), it is clear that $L$ and $N$ are not isomorph, but $M/N\cong (x^k, k\geq n+1)\cong (x^k,k\geq n+2)\cong M/L$.
$(x^k, k\geq n+1)$ the vector subspace generated by $x^k, k\geq n+1$.