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Part CCSIR NET December 2023

Ideals, quotient rings, prime & maximal ideals, CRT

Let be such that n ≡ 1 (mod 7) and n ≡ 4 (mod 15). Which of the following statements are true?

  1. A.n ≡ 1 (mod 3).✓
  2. B.n ≡ 1 (mod 35).
  3. C.n ≡ 1 (mod 21).✓
  4. D.n ≡ 1 (mod 5).

Solution

n ≡ 4 (mod 15) gives n ≡ 1 (mod 3) and n ≡ 4 (mod 5). With n ≡ 1 (mod 7), CRT gives n ≡ 1 (mod 21); n ≢ 1 (mod 5) kills the mod-35 claim.

Related counterexample: Every prime ideal is maximal

More on this topic

From Rings and Fields › Ideals, quotient rings, prime & maximal ideals, CRT

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