Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

How prove that the ideal $(x)=x\mathbb{Z}[x]$ in $\mathbb{Z}[x]$ is prime but not maximal?

share|improve this question
Low accept rate discourage some from answering your posts. When a question has been answered to your satisfaction, it's considered good form to click the check mark below the voting arrows for that answer. The person whose answer you accept (you can only accept one) gets a small reputation boost, and you send a signal to others that you're no longer in need of an answer. BTW, it's not too late to go back to your old posts and accept your favorite answers. –  draks ... Jul 2 '12 at 8:06

4 Answers 4

$p(x).q(x)\in (x)\Rightarrow p(0).q(0)=0\Rightarrow p(0)=0\mbox{ or }q(0)=0\Rightarrow x|p(x)\mbox{ or }x|q(x)$ hence prime ideal

As $(x)\subset(x)+(2)\subset\mathbb{Z}[x]$ hence not maximal

share|improve this answer

Show that $\mathbb{Z}[x]/(x)$ is an integral domain but not a field.

share|improve this answer

Zooming out for a moment: the following general facts should help. Let $A, B$ be commutative rings.

  1. An ideal $\mathfrak a$ of $A$ is prime (resp. maximal) if and only if $A/\mathfrak a$ is an integral domain (resp. a field).
  2. There is a canonical isomorphism from $B[x]/(x)$ to another ring. You probably have a guess for what the ring and the map are and it's likely correct; to prove it, try to define an inverse.

Related, but possibly overwhelming at this point: Mumford's treasure map.

share|improve this answer

Structurally: use $\rm\:P\:$ is prime $\rm\iff R/P\:$ is a domain; $\rm\:P\:$ is maximal $\rm\iff R/P\:$ is a field.

Elementwise: $\rm\ \ f,g\not\in (x)\iff f(0),g(0)\neq 0\iff f(0)g(0)\neq 0 \iff fg\not\in (x)$

$\rm (x)$ max $\rm\iff [f\not\in(x)\, \Rightarrow\, (f,x)=1]\iff [f(0)\ne 0\:\Rightarrow (f(0)) = 1]\,$ by evaluation at $\rm\,x=0.$

It is very instructive to analyze how the elementwise proofs correspond to the structural proofs.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.