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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
3 years, 6 months 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. · 3 years, 6 months ago