Galois connection for annhilators https://en.m.wikipedia.org/wiki/Annihilator_(ring_theory)
Under the section "Category-theoretic description for commutative rings"
Wikipedia states few results for galois connection for annihilator for modules.
$$1. ann(ann(ann(S)))=S$$ and
$$2.span(S)\subset ann(ann(S))$$.
It makes me wonder if such results exists for annhilators of ideals in the ring.
I tried few easy examples
$span(S)\subset ann(ann(S))$ this holds in any integral domain. $ann(ann(ann(S)))=S$ this too holds.

I wonder if these two results is true in general. ( I doubt 1 maynot be hold)


Also if $span(S)\subset ann(ann(S))$ holds in general, under what condition they are equal.

One condition I found is if $span(s)$ is generated by an idempotent  then equality holds but I can't think anymore general.
 A: I think you must have typoed the first one. I think you rather mean

$ann(ann(ann(S))) = ann(S)$

because, for example, if you take the zero ideal in $\mathbb Z$, your first equation is wrong as written.
I'm going to use some ad hoc notation to write the general case.
Let $M$ be a right $R$ module and write
$ann(X)=\{r\in R\mid Xr=\{0\}\}$ for a nonempty subset $X\subseteq M$
$Ann(Y)=\{m\in M\mid mY=\{0\}\}$ for a nonempty subset $Y\subseteq R$
Now note:

*

*Just by the definitions of annihilators, $Y\subseteq ann(Ann(Y))$ and $X\subseteq Ann(ann(X))$.


*Furthermore, it is easy to check that both maps $ann()$ and $Ann$ are containment reversing.
Now, using the first equation, substituting $ann(X)$ for $Y$, you'd get $ann(X)\subseteq ann(Ann(ann(X)))$. On the other hand, applying $ann$ to both sides of $X\subseteq Ann(ann(X))$, you get $ann(X)\supseteq ann(Ann(ann(X))$.  Therefore $ann(X)=ann(Ann(ann(X)))$.
Using a similar argument, you have $Ann(Y)=Ann(ann(Ann(Y)))$.
From this, you can derive that the left and right annihilator maps in $R$ satisfy $\ell r \ell = \ell$ and $r \ell r=r$, and in the case of a commutative ring you'd have the identity you proposed above: $ann^3=ann$.

As for the second part, from $Y\subseteq ann(Ann(Y))$ you have that the left ideal generated by $Y$ is contained in $ann(Ann(Y))$ since $ann(Ann(Y))$ is a left ideal of $R$.
And likewise, you have that the submodule generated by $X$ is contained in $Ann(ann(X))$ since $Ann(ann(X))$ is a submodule of $M$.
But in general it does not have to be the case that you have equality.
You can work out why $ann(Ann(L))=L$ if and only if $L$ is of the form $ann(N)$ for some submodule $N$ of $M$.
Likewise $Ann(ann(N))=N$ if and only if $N=Ann(L)$ for some left ideal $L$ of $R$.
When $M$ is the ring itself, ideals like these are called "annihilator ideals" or sometimes "annulets" but I am not completely sure what they call them in the general case when looking at the annihilator maps between a module and its ring.
