If $I$ is an ideal in a ring $R$ let $[R:I] =\{r \in R\mid xr \in I\text{ for every }x \in R\}$. How can I show that $[R:I]$ is an ideal of $R$ which contains $I$.
Prove that $[R:I] =\{r \in R\mid xr \in I\text{ for every }x \in R\}$ is an ideal of $R$ that contains $I$
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abstract-algebra
ring-theory
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0You have swapped the "numerator" and the "denominator" in your definition of what is sometimes called a colon ideal. The correct notation is $ (I:R) =\{r \in R\mid xr \in I\text{ for every }x \in R\} $ – 2012-04-27
1 Answers
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Ideal: just check directly using the definition of ideal.
Containment: You need to show that elements of $I$ have the necessary property, ie. for all $x\in I$, $xr\in I$ for each $x\in R$.
Both follow directly from the definitions: once you grok the definition of Ideal and $[R:I]$ it should be pretty quick.
Bonus question: Show that $I=[R:I]$ if $R$ contains $1$.
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0Suppose that x and y belong to [R:I]. Then for every r in R, rx and ry belong to R, so r(x-y) belong to R, and hence x-y belongs to [R:I]. This shows that [R:I] is a group. Further, if x belongs to [R:I] and r, r' belong to r, then r'(rx) = (r'r)x belongs to [R:I]. This shows that [R:I] is an ideal. This is a quick idea think I'm on the right path? – 2012-04-27