Part CCSIR NET December 2024boolean-rings-can-be-domains

Boolean rings can be domains

Let R be a nonzero ring with unity such that $r^{2} = r$ for all r ∈ R. Which of the following statements are true?

A.R is never an integral domain.
B.r = −r for all r ∈ R.✓
C.Every nonzero prime ideal of R is maximal.✓
D.R must be a commutative ring.✓

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

50 are analysed free — try those first.

Related counterexample: Every prime ideal is maximal

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The chapter behind this: Ideals, quotients and the Chinese remainder theorem — free to read

From Rings and Fields › Ideals, quotient rings, prime & maximal ideals, CRT

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