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Part BCSIR NET December 2024over-the-complexes-a-sum-of-squares-factors

Over the complexes a sum of squares factors

Let C[x,y]\mathbb{C}[x, y] be the polynomial ring in two variables over C\mathbb{C}. For which of the following ideals I, the quotient ring C[x,y]/I\mathbb{C}[x, y]/I is NOT an integral domain?

  1. A.I = (x, y)
  2. B.I = (x + y)
  3. C.I=(x2+y2)I = (x^{2} + y^{2})✓ Correct
  4. D.I = (xy − 1)

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 base field or ring.

See pricing

50 are analysed free — try those first.

The trap it tests

Base field or ring

The answer changes with the field or ring you are working over.

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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