For which Noetherian local rings, is every associated prime ideal contained in the square of the maximal ideal

Question

Let $(R,\mathfrak m, k)$ be a Noetherian local ring of depth $>1$. If $a\in R$ is a zero-divisor, i.e. if $ab=0$ for some $0\ne b \in R$, then must it be true that $a\in \mathfrak m^2$? Since the collection of all zero divisors is the union of all associated primes, so in other words I'm asking: Is it true that every associated prime is inside $\mathfrak m^2$?

Answer 1

Try $R=k[[x,y,z]]/xy$. It has depth two, $xy=0$, but $x$ is not in the square of the maximal ideal.