Let $R$ be a ring and $N$ be an $R$-submodule of $M$. For the element $0\neq a\in R$, under what conditions can the following hold: $$Na=Ma\cap N~?$$
I already know that $Na\subseteq Ma$ and $Na\subseteq N$ which implies that $Na\subseteq Ma\cap N$. So I only require cases when $Ma\cap N\subseteq Na$.