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
·
3 years, 3 months ago

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TopNewestI 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 · 3 years, 3 months ago

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– Insan Budiman Mahdar · 3 years, 3 months ago

thx sir. :)Log in to reply