1

Consider the ring $\Bbb Z[x]$.

Does there exist a maximal ideal $M$ such that $\Bbb Z[x] /M \simeq \Bbb Q$ ?

I don't think so, because $\Bbb Z[x]$ is not field.

Olivier Bégassat
  • 20,248
heeju
  • 23
  • 5
    In the future, please use $\LaTeX$ in your questions. As for the question itself, your reasoning is flawed; $\mathbb Q[x]$ is not a field, but there is a maximal ideal (namely $(x)$) such that $\mathbb Q[x]/(x) \simeq \mathbb Q$. In fact if $\mathbb Z[x]$ were a field, there would only be one maximal ideal: $(0)$. – Najib Idrissi Aug 03 '13 at 09:24
  • 1
    The phrasing of the original form of this question did leave a lot to be desired. But I think that criticizing new users for not using LaTeX is misguided. – Jyrki Lahtonen Aug 03 '13 at 09:40

1 Answers1

3

Such a maximal ideal $M$ cannot exist. This is because $\mathbb{Z}[x]$ is generated as a ring by $x$. Therefore $\mathbb{Z}[x]/M$ is generated as a ring by the coset $x+M$. But $\mathbb{Q}$ is not generated as a ring by any single element. This is because the subring generated by $q=m/n$ only has elements with denominators that are factors of powers of $n$.

Jyrki Lahtonen
  • 133,153