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Part CCSIR NET June 2024irreducible-quotient-need-not-be-a-ufd

Irreducible quotient need not be a ufd

For two indeterminates x, y, let R = 𝔽3[x]_{3}[x] and S = R[y]. Which of the following statements are true?

  1. A.S is a principal ideal domain
  2. B.S/(y2+x2)S/(y^{2} + x^{2}) is a unique factorization domain
  3. C.S is a unique factorization domainβœ“ Correct
  4. D.S/(x) is a principal ideal domainβœ“ Correct

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times β€” the specific reasoning error each wrong option was built to reward, and why this one tests property not inherited.

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50 are analysed free β€” try those first.

The trap it tests

Property not inherited

A property assumed to pass to subobjects, quotients, or through a chain. It does not.

Drill statements like this

Related counterexample: Irreducible over β„€ β‡’ irreducible mod every prime

More on this topic

The chapter behind this: Irreducibility tests β€” free to read

From Rings and Fields β€Ί Polynomial rings and irreducibility tests

Last revised . Found a mistake? Tell us β€” corrections are the fastest thing we act on.

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