Let R be a commutative ring. If I and P are idelas of R with P prime such that I !sube P, prove that the ideal P:I=P

Maiclubk 2020-11-27 Answered
Let R be a commutative ring. If I and P are idelas of R with P prime such that I!P, prove that the ideal P:I=P
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Expert Answer

Maciej Morrow
Answered 2020-11-28 Author has 98 answers

The ideal quotient of PandI is P:I={xR:xIP} which is again an ideal of R.
Given that P is a prime ideal.
To prove that P:I=P
We prove this set equivalence by proving that one set is the subset of other.
For, let xP
Then xIP since P is an ideal.
Therefore, PP:I
Conversely, let xP:I
Then xIP. That is xyP , for every yI.
Since I!P, there exists yI such that yI. But xyP.
Also since P is a prime ideal, x must be P.
Hence xP which implies that PP:I
Hence proved.

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