Given a ring $R$ and an ideal(two-sided) $I\subset R$, we find an ideal $$[R:I]=\{x\in R| xR\subset I \}$$
It is easy to see that this ideal coincides with the original ideal $I$ if $I$ is a prime ideal. As I can see such ideals give an extension of the original ideal and satisfy an order-reversing property, i.e. $$J\subset I \Rightarrow [R:I]\subset [R:J]$$.
I would like to know more about this ring. I checked the list of ideals on Wikipedia, but it was not helpful.
Secondly, is there a name also for the following ideal? The best I can think of is "annihilator of $I$ in $R$". $$r(I)=\{r\in R| rI=\{0\}\}$$
Thanks