Suppose that $R$ is a (local) ring and $r\in R$. When do the equations $Ann_R(Ann_R(r))=Rr$ or $\sqrt{Ann_R(Ann_R(r))}=\sqrt{Rr}$ hold? I already know that it holds for Artinian Gorenstein rings (due to an exercise in BrunsHerzog) and it seems to be true for $R=\Bbb {Z}/n\Bbb {Z}$. The question is more interesting when we also assume that $Ann_R(r)\in Ass(R)$

For what it's worth, it suffices for $(r)$ to be an interesection of minimal primes; in fact, more generally if $J$ is any intersection of minimal primes then $J=Annih(Annih(J))$. (This requires only that $R$ be commutative and noetherian; you don't need local.) This is Lemma 2.17 in an old paper of mine called "Patching Modules of Finite Projective Dimension", where the stated hypothesis is "Let $R$ be any (commutative) ring", but it now seems to me that the proof requires $R$ to be noetherian. 


Let $l(X)$ denote a left annihilator, and $r(X)$ denote a right annihilator. A ring is called right Pinjective if $l(r(a))=Ra$ for all $a\in R$. You can find this condition discussed in detail in Nicholson and Yousif's QuasiFrobenius rings on page 96. Rings which are right selfinjective are right $P$injective (and so that's why $\Bbb Z/n\Bbb Z$ has the property for every $n>1$.) Also, all von Neumann regular rings are left and right $P$ injective, and these can be noncommutative, nonNoetherian and nonselfinjective. 


Long time ago, I proved, but never published, that if $R$ is generically Gorenstein, meaning that all $R_P$ are Gorenstein with $P$ an associated (necessarily) minimal prime, then $\operatorname{Ann}(\operatorname{Ann}(I))=I$, for any ideal $I$ with nonzero annihilator. 

