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

Boolean rings can be domains

Let R be a nonzero ring with unity such that r2=rr^{2} = r for all r ∈ R. Which of the following statements are true?

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

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, and why this one tests standard counterexample.

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The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

Related counterexample: Every prime ideal is maximal

More on this topic

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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