GATE 2023 MA – Question 45
Let R={p(x)∈ℚ[x]:p(0)∈ℤ}. For a∈R write (a) for its generated ideal. Which statements hold?
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: (A) If p is irreducible in R then (p) is a prime ideal.
Explanation
The official key selects **A only**, which is retained for grading. However D is also mathematically true for the printed ring: (x)⊊(x/2)⊊(x/4)⊊… is a strictly increasing chain of ideals, so R is non-Noetherian and cannot be a PID. B fails because x can be divided by every power of 2 within R and has no finite irreducible factorization. C fails since x/2∉(x), but (x/2)²=x(x/4)∈(x). For A, irreducibles are integer primes or rational-polynomial irreducibles with constant term ±1 (up to units). The first give quotient ℤ/pℤ; the latter give the field ℚ[x]/(p), since their nonzero constant term makes x invertible modulo p and permits rational constants. Both yield prime ideals. Thus this source/key discrepancy is explicitly disclosed.