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 Bruns-Herzog) and it seems to be true for $R=\Bbb {Z}/n\Bbb {Z}$.
The question is more interesting when we also assume that $Ann\_R(r)\in Ass(R)$



When does "second annihilator" of a (principal) ideal equal the ideal itself , ie $Ann_R(Ann_R(r))=Rr$?
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.


