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Part CCSIR NET June 2024crt-splits-the-ideal-lattice

Crt splits the ideal lattice

Let I be an ideal of the ring 𝔽2[t]/(t2(1−t)2)_{2}[t]/(t^{2}(1 - t)^{2}). Which of the following are the possible values for the cardinality of I?

  1. A.1✓ Correct
  2. B.8✓ Correct
  3. C.16✓ Correct
  4. D.24

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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