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Part CCSIR NET December 2025z-mod-9-satisfies-the-chain-condition-and-still-has-zero-divisors

Z mod 9 satisfies the chain condition and still has zero divisors

Consider a commutative ring R with unity with at least five elements such that for any two elements a,b∈R, there exists c∈R such that a=bc or b=ac. Which of the following statements are necessarily true?

  1. A.R has a unique maximal ideal.✓
  2. B.Every finitely generated ideal of R is principal.✓
  3. C.R is an integral domain.
  4. D.R is a euclidean domain.

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.

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50 are analysed free — try those first.

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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