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What are the prime ideals of $\mathbb{R}[x_1,x_2,x_3,...]$?

(this is the ring of polynomials over the reals with countably infinite many indeteminates).

My attempt: I think taking the principal ideal generated by an irreducible polynomial gives a prime ideal.

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    I would say your attempt is correct, but do notice that $\langle 0 \rangle$ is also prime (and is the only one that you can't reach in this way2012-06-10
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    @Belgi: But why are these the only prime ideals?2012-06-10
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    In second thought I think that what I said only applies to the polynimial ring in one variable (over a *field*). I think that your ring is not a PID...2012-06-10
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    @Belgi: Pretty sure it's not a PID, but maybe all prime ideal are principal.2012-06-10
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    No, they aren't. For example $(x_1, x_2)$ is prime, as $\mathbb R[x_1, x_2, x_3, \ldots]/(x_1, x_2) \cong \mathbb R[x_3, x_4, \ldots]$ is integral.2012-06-10
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    @martini: I see. But I'm not sure how to generalize to the full answer.2012-06-10
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    This is homework? It seems to me that it is quite hard to give a reasonable description of all prime ideals of this ring. Here's one that isn't finitely generated: $(x_1^2 - x_2, x_2^2 - x_3, x_3^2 - x_4, ...)$. Are you sure the question isn't about _maximal_ ideals?2012-06-10
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    @QiaochuYuan: This is actually inspired by homework and not exactly homework. Sorry for misleading.2012-06-10
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    @QiaochuYuan: So, taking the quotient by the ideal you presented gives $\mathbb{R}[x]$?2012-06-10
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    @Ola: yes. Okay, so in that case I do not think there is a reasonable description of all prime ideals of this ring. Already I do not think there is a reasonable description of all prime ideals of $\mathbb{R}[x_1, x_2, x_3]$ (and killing the variables $x_n, n \ge 4$ shows that this problem is a subproblem of your problem).2012-06-10
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    @QiaochuYuan: Is there an example of a non-noetherian ring for which we can give an easy description of all prime ideals?2012-06-10
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    @Ola: here is an example of easy description. The non-noetherian ring $\mathbb R[x_1,x_2,..]/(x_i\cdot x_j\mid i,j=1,2,3,...)$ has $(x_1,x_2,x_3,...)$ as its only prime ideal.2012-06-10
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    @GeorgesElencwajg: Isn't this ideal equal the entire ring?2012-06-10
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    @Ola: no. It only contains the elements of positive degree so doesn't contain the constants.2012-06-10
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    @QiaochuYuan: Oh, I see. I should think about why it's prime and why there are no other primes. Thanks for the help.2012-06-10

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