The GATE Grind

GATE Algebra: Rings and Ideals – Previous Year Questions

11 GATE previous year questions on Rings and Ideals (Algebra, Mathematics) with answers and explanations, from every paper.

  1. GATE 2021 MA Q28 (1 mark, Numerical answer) – Let I=(x²+x+1) in R=ℤ₃[x]. Find the number of units in R/I.
  2. GATE 2021 MA Q36 (2 marks, Multiple choice) – In R=a+b√(−17):a,b∈ℤ, P:2+√(−17) is irreducible. Q:2+√(−17) is prime. Choose truth values.
  3. GATE 2022 MA Q36 (2 marks, Multiple choice) – Which describes ℝ[X]/(X⁴+4)?
  4. GATE 2022 MA Q37 (2 marks, Multiple choice) – For proper nonzero ideals J₁,J₂ of a nonzero commutative ring R, P:for every r₁,r₂∈R there is a unique r∈R with r−r₁∈J₁,r−r₂∈J₂. Q:J₁+J₂=R. Which…
  5. GATE 2023 MA Q45 (2 marks, Multiple select) – Let R=p(x)∈ℚ[x]:p(0)∈ℤ. For a∈R write (a) for its generated ideal. Which statements hold?
  6. GATE 2023 MA Q46 (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?
  7. GATE 2024 MA Q15 (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.
  8. GATE 2025 MA Q59 (2 marks, Multiple select) – Rings are commutative associative with 1≠0 and unital homomorphisms. P1:R is a product of two nonzero rings. P2:R has nonzero idempotents r₁,r₂ with…
  9. GATE 2026 MA Q23 (1 mark, Multiple select) – Over ℤ₁₁, let p(x) = x²+x+4. Which statements are true?
  10. GATE 2026 MA Q41 (2 marks, Multiple choice) – Which ring under usual addition and multiplication is a unique factorization domain?
  11. GATE 2026 MA Q52 (2 marks, Multiple select) – T: Q[x] Q[x] sends p(x) to p(x²). Which statements hold?