give an example of an ideal ring R and A and B so that AB ⊂ A ∩ B
Insan Budiman Mahdar
4 years ago
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.
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thx sir. :)