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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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    Your $r(I)$ is simply $\operatorname{Ann}(I)$, the annihilator of $I$ when $I$ is viewed as an $R$-module.2012-03-10
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    In addition to "ideal quotient", I've heard this called a "colon ideal".2012-03-10
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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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    Thank you. I will have to prove the second remark.2012-03-10
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    I think I have a counterexample $[4\mathbb{Z}:2\mathbb{Z}] = 2\mathbb{Z}$. Your second comment works when $R$ is a ring with unity. Is that right?2012-03-10
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    If $x \in [I : R]$, then $x = x \cdot 1 \in xR \subset I$.2012-03-10
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    Oh, right. I usually assume people mean ring with unity unless they say otherwise.2012-03-10