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Part BCSIR NET June 2025a-valuation-ring-of-Q-is-local-with-only-two-prime-ideals

A valuation ring of Q is local with only two prime ideals

Let A be a subring of the field of rationals Q\mathbb{Q} such that for any nonzero rational r∈Q,r∈Ar \in \mathbb{Q}, r \in A or 1/r ∈ A. Which of the following statements is FALSE?

  1. A.The set {a ∈ A : 1/a ∉ A} ∪ {0} is an additive subgroup of Q\mathbb{Q}.
  2. B.A has at most one maximal ideal.
  3. C.If A≠QA \ne \mathbb{Q}, then A has infinitely many prime ideals.✓ Correct
  4. D.For any nonzero a, b ∈ A, a divides b or b divides a in A.

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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50 are analysed free — try those first.

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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