Isomorphic quotient modules implies equal submodules? Let $R$ be a commutative ring with unity $M$ an $R$-module and $N,L$ submodules of $M$ with $N\subseteq L$.
$$M/N\cong M/L\implies N=L\ ?$$

No, this is not true. Consider (the $\Bbb Z$-module) $M=\Bbb Z\times\Bbb Z\times \cdots$, and the two submodules $N=\langle(1,0,0,\ldots)\rangle$ and $L=\langle(1,0,0,\ldots), (0,1,0,0,\ldots)\rangle$. Then $M\cong M/N\cong M/L$, but $N$ and $L$ are not only unequal, they aren't even isomorphic.
