Let $R$ be a commutative ring and $N$ be a submodule of an $R$-module $M$ such that for each $a\in R$, $$aN=aM\cap N~~\text{and}~~M/aM\cong (0:_Ma)$$ where $(0:_Ma)=\{m\in M|am=0\}$ a submodule of $M$. Is there any such isomorphism like $$(M\cap N)/(aM\cap N)\cong (0:_Ma)\cap N ?$$
I was using the Second Isomorphism Theorem and the modular law to examine $N/(aM\cap N)$ and I obtained the following: $$N/(aM\cap N)\cong (N+aM)/aM=(M\cap N+aM)/aM\cong (M\cap N)/(M\cap aM\cap N)=(M\cap N)/(aM\cap N).$$ Now the last expression is tempting me to conclude that $$(M\cap N)/(aM\cap N)\cong (0:_Ma)\cap N!$$ since $M/aM\cong (0:_Ma)$. I need some guidance whether this is wrong or right.