When is the intersection of every non-zero ideal with socle again non-zero?

Question

Let $(R,\mathfrak m)$ be a Noetherian local ring such that $(0:_R \mathfrak m)\neq 0$. Then, is it true that for every non-zero ideal $I$ of $R$, one also has $I\cap (0:_R \mathfrak m)\neq 0$ ?

I know this is true when $R$ is Artinian, but not sure otherwise.

Please help.