When an ideal contains an annihilator For a commutative ring $R$  with 1, it is easily seen that $I\subseteq ann_R (J)$ if and only if $IJ=0$, where $I$ and $J$ are two arbitrary  ideals of $R$. I am looking for an equivalence condition for $ann_R(I)\subseteq J$. 

Note that $ann_R (A)=\{r\in R \mid rA=0\}$ for any set $A\subseteq R $.

 A: Although the two expressions ($I\subseteq ann_R(J)$ and $ann_R(I)\subseteq J$) appear very analogous, their flavors are quite different.
You can interpret $ann_R(J)$ lattice theoretically and find that it has two important properties:


*

*It is the largest element of the poset of ideals $\{I\lhd R\mid IJ=\{0\}\}$; and 

*this poset is downward closed. 


These two properties (that there is a largest element, and that the elements beneath it all have the sam properties) allows the "characterization" that $I\subseteq ann_R(J)\iff IJ=\{0\}$.
On the other hand, the properties of the annihilator don't entail conclusions about things "above" the annihilator.
The only relevant extension I can think of for this relies on special ideals, namely prime ideals. The hull-complement of an ideal $I$, write it as $h^c(I)$,  is the set of prime ideals of $R$ not containing $I$. It's obvious that $ann_R(I)\subseteq \bigcap h^c(I)$, so that is one interesting conclusion we can draw, and a source of an upward-closed set of candidates for $J$.
The intersection of the hull-complement and any ideals above it are all always among the $J$'s such that $ann_R(I)\subseteq J$. (But obviously $ann_R(I)$ itself is among those ideals too.) It would be interesting to know when this is minimal, so that there are no ideals between $ann_R(I)$ and $\bigcap h^c(I)$, or if no ideals fell outside either category.
