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

Unit generates the whole ring

Consider the ring R = { $\sum_{(} n \in Z) a_{n} X^{n}$ | $a_{n} \in Z$ , and $a_{n} \neq = 0$ only for finitely many $n \in Z$ }, with the usual addition and multiplication of such sums. Which of the following statements is true?

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

See pricing

50 are analysed free — try those first.

Related counterexample: Every prime ideal is maximal

Unit generates the whole ring

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