A question about the injective hull of the residue field

Question

Let $(R, {\frak m})$ be a local ring and set $k=R/{\frak m}$. Over a Gorenstein local ring of dimension $n \geq 1$ we have $E_R(k)={\frak m} \, E_R(k)$. Is the relation true when $\dim R \geq 1$ and $R$ is not necessarily Gorenstein?