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GATE 2023 MA – Question 46

Algebra · Rings and Ideals · 2 marks · Multiple select

Let S₁=ℤ[x]/(2,x³) and S₂=ℤ₂[x]/(x²), with the displayed ideals generated by their entries. Which statements hold?

  1. Every prime ideal of S₁ is maximal.
  2. S₂ has exactly one maximal ideal.
  3. Every element of S₁ is nilpotent or a unit.
  4. S₂ has an element that is neither nilpotent nor a unit.

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Show answer and explanation

Correct answer: (A) Every prime ideal of S₁ is maximal.; (B) S₂ has exactly one maximal ideal.; (C) Every element of S₁ is nilpotent or a unit.

Explanation

S₁≅𝔽₂[x]/(x³), S₂≅𝔽₂[x]/(x²). Each has only one prime ideal, generated by x, with field quotient 𝔽₂. A class with zero constant coefficient is nilpotent; one with constant 1 has a finite geometric-series inverse. Thus A,B,C hold and D fails. **A,B,C**.

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