Testing for ideals

Question

Let J be the ideal of a ring $\mathbb{Q}[x]$ generated by monomial $x$.

i) Is J a prime ideal?

ii) Is it a maximal ideal?

iii) Describe the quotient ring $\mathbb{Q}/J$

Okay so for i) I said Yes, ex. Let a and b be two polynomials in $Q[x]$ then $x\mid ab \implies x\mid a \text{ or }x\mid b $, hence it is a prime ideal (not sure if correct) But this was my thought.

ii) not sure on how to deal with this one since we're dealing with a polynomial.

iii) I would say the quotient ring is $\mathbb Q[x]/(x)$ where $(x)$ is the principal ideal geberated by $x$ (still not sure if this is correct)...

Answer 1

Actually, there is a surjective ring homomorphism \begin{align} \mathbb Q[x]&\longrightarrow\mathbb Q,\\p(x)&\longmapsto p(0). \end{align} Its kernel is precisely the ideal $(x)$, hence the first isomorphism theorem yields an isomorphism $$\mathbb Q[x]/(x)\simeq\mathbb Q,$$ which proves both the ideal $(x)$ is maximal and it is prime, and describes the quotient ring..

Answer 2

If $ Q $ has zero divisors then I) is wrong as x does not divide constant polynomials but it will divide their product if it happens to be $0 $.

Answer 3

Hint: There is a much simpler description of $\mathbb{Q}[x]/(x)$ than the one you have given. Figure out what ring that is isomorphic to. Once you do that, the answers to (i) and (ii) should follow from the fact that an ideal is prime (maximal) if and only if the quotient ring is a domain (field).