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Part BCSIR NET June 2024unit-generates-the-whole-ring

Unit generates the whole ring

Consider the ring R = { ∑(n∈Z)anXn\sum_(n\in\mathbb{Z}) a_{n}X^{n} | an∈Za_{n} \in \mathbb{Z}, and an≠0a_{n} \ne 0 only for finitely many n∈Zn \in \mathbb{Z} }, with the usual addition and multiplication of such sums. Which of the following statements is true?

  1. A.R is not commutative
  2. B.The ideal (X − 1) is a maximal ideal in R
  3. C.The ideal (X − 1, 2) is a prime ideal in R✓ Correct
  4. D.The ideal (X, 5) is a maximal ideal in R

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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