Local ring and zero divisors 
Let $(R,\mathfrak m)$ be a Noetherian local ring. Suppose that all $x$ belonging to $\mathfrak m-\mathfrak m^2$ are zero divisors. Show that all elements of $\mathfrak m$ are zero divisors.

My progress:
My idea is to see that the set of zero divisors is equal to the union of the associated primes of the ideal zero, obtained from the minimal primary decomposition of the ideal zero.

In fact, under the given hypothesis there is a nonzero element $x$ of $R$ with
$x \mathfrak{m} = \{0\}$.  This is proved in Lemma 5.85 of T.Y. Lam's GTM Lectures on Modules and Rings.  Lam's proof is reproduced in Corollary 10.8 of these notes.
