Part CCSIR NET June 2025injective-and-surjective-part-company-once-the-ring-is-infinite-dimensional

Injective and surjective part company once the ring is infinite dimensional

Let $f : Q [X] \to Q [X]$ be a ring homomorphism with f(1) = 1. For n ≥ 1, let $f^{n} = f$ ∘ ⋯ ∘ f (n times). Which of the following statements are true?

A.If f is onto, then so is $f^{n}$ for all n ≥ 1.✓
B.ker $f^{n + 1}$ $= ker f^{n}$ for some n ≥ 1.✓
C.If f is onto, then f is one-to-one.✓
D.If f is one-to-one, then f is onto.

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 finite-dimensional intuition.

See pricing

50 are analysed free — try those first.

The trap it tests

Finite-dimensional intuition

Something true in ℝⁿ, or in a nice space, assumed in general.

Drill statements like this

Related counterexample: Irreducible over ℤ ⇒ irreducible mod every prime

Injective and surjective part company once the ring is infinite dimensional

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