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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

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    BrianM.Scott and @Hurkyl, thank you.$I$have also found colon ideals on Wikipedia.2012-03-10

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What you denoted by $[R : I]$ is called an ideal quotient. Three points here: first, the ordering you used is nonstandard, one usually writes $[I : R]$ for that ideal. Second, $[I : R] = I$ always (when $R$ is unital). Third, to make this interesting take another ideal $J \subset R$ and consider $[I : J] = \{ x \in R \ | \ xJ \subset I \}$. This is a general ideal quotient.

Yes, $r(I)$ is called the annihilator of $I$.

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    Oh, right. I usually assume people mea$n$ ring with unity unless they say otherwise.2012-03-10