Let $A$ be a commutative regular local ring of dimension $d$ with maximal ideal $\mathcal m$ and $a \in A$ an element of the ring.
Suppose that $\mathcal m \cdot a \subset \mathcal m^2$, i.e. if I multiply the element $a$ by an arbitrary element of $\mathcal m$, then I am in the square ideal of $\mathcal m$.
Can I conclude from this that already $a \in \mathcal m$?