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Aljabar

give an example of an ideal ring R and A and B so that AB ⊂ A ∩ B

Note by Insan Budiman Mahdar
4 years ago

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I presume you want \(R\) to be a ring and \(A,B\) to be ideals.

Try letting \(R\) be the collection of subsets of some universal set \(X\). This forms a ring with addition being "exclusive or": \[ U + V \; = \; (U \cap V') \cup (U' \cap V) \qquad U,V \in R \] and multiplication being intersection: \[ U \times V \; = \; U \cap V \qquad U,V \in R \] Think what the principal ideals of \(R\) must look like. You can get your example by considering principal ideals only.

Mark Hennings - 4 years ago

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thx sir. :)

Insan Budiman Mahdar - 4 years ago

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