GATE 2024 MA – Question 15

Algebra · Rings and Ideals · 1 mark · Multiple choice

In the subring ℝ[X²,X³] of ℝ[X], I: the ring is a UFD; II: it is a PID. Choose truth values.

Both I and II true
I true, II false
I false, II true
Both false

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Correct answer: (D) Both false

Explanation

Its nonconstant monomials X² and X³ are irreducible because no polynomial of degree one belongs to the ring. But X⁶=(X²)³=(X³)² gives inequivalent factorizations; X² and X³ are not associates. Thus it is not a UFD. Every PID is a UFD, so it is not a PID either. **D**.

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