Question

Let ℤ[𝑥] be the polynomial ring with coefficients in ℤ. Prove or disprove 

that the ideal 𝐼 = (4, 𝑥) is a principal ideal in ℤ[𝑥]. Is the ideal 𝐼 a 

maximal ideal in ℤ[𝑥]? Explain your answer.

Answer 1

The ideal 𝐼 = (4, 𝑥), by the definition, consists of those polynomials over Z, which can be expressed as a sum $4\cdot u(x) + x\cdot v(x)$.

Every such polynomial f(x) satisfies the condition $f(0)\equiv 0\ (mod\ 4)$. Therefore, $1\notin I, 2\notin I$.

If we assume that I is a principal ideal in ℤ[𝑥], then there exists a polynomial d(x) in I, which divides all elements in I.

d(x) is a divisor of 4. All divisors of 4 are: 1, 2, 4, -1, -2, -4. Only 4 and -4 are elements of I, but no one of them is a divisor of x.

The contradiction proves that I is not a principal ideal.

Since $2\notin I$ , the ideal I' = (2, x) is greater than I. Moreover, $1\notin I'$, because of each polynomial f(x) from I' satisfies the condition $f(0)\equiv 0\ (mod\ 2)$.

Therefore, I is not a maximal ideal.