Prove that in $\mathbb{Z}[X]$ the ideal generated by $X$, i.e. $I=\langle X\rangle$, is a maximal principal ideal (that is, maximal among principal ideals), but is not a maximal ideal.
Maximal principal ideal which is not maximal
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$\begingroup$
abstract-algebra
ring-theory
ideals
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1$I=\langle X\rangle \subsetneq \langle 2,X\rangle$, so it is not maximal. – 2012-10-28
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2Why post in the imperative? Are you assigning homework to us? – 2012-10-28
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0No, I'm sorry, I did not intend that. – 2012-10-28
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0but why $
$ is a maximal ideal principal of $\mathbb{Z}[X]$ – 2012-10-28 -
0I've never heard the phrase "maximal ideal principal". Could you define it? – 2012-10-28
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0I think it's [romance language](https://en.wikipedia.org/wiki/Romance_language) word order for "maximal principal ideal", i.e. ideal which is maximal among principal ideals. – 2012-10-28
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0I mean in all principal ideal of $\mathbb{Z}[X]$ then $
$ is maximal. OK? – 2012-10-28 -
0I think if $
$ is another principal ideal of $\mathbb{Z}[X]$ then $ – 2012-10-28\subset $. This mean if $f(X)\in $ then exits $g(X)$ s.t $f(X)=g(X).h(X)$, then what is $k(X)$ s.t $f(X)=x.k(X)$?. -
1Could you please use `\langle X\rangle` instead of `
`. I'm sure you can see the spacing is all wrong and it makes it *really* hard to read. – 2012-10-28
1 Answers
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$I$ is not maximal because it's contained in $\langle 2,X\rangle$, as Sigur noticed, which is an ideal which stricly contains $I$ and is itself strict.
But it's maximal among principal ideals. Indeed, let $I'$ a principal ideal containing $I$, say generated by $P_0$. If $P\in I'\setminus I$, we have $P(0)\neq 0$ (otherwise $P\in I$). Write $P:=\underbrace{\sum_{j=1}^Na_jX^j}_{\in I\subset I'}+a_0$, then $a_0\in I'$. As $a_0=P_0Q_0$ for some $Q_0\in\Bbb Z[X]$, taking the degrees on both sided, $P_0$ is constant so $I'=\Bbb Z[X]$.
Conclusion: the only principal ideal containing $I$ is $\Bbb Z[X]$.
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0So, $\langle X\rangle \subset \langle 1\rangle$, and $\langle 1\rangle$ indeed principal ideal. Are we convention a ideal of $\mathbb{Z}[X]$ have to different $\mathbb{Z}[X]$?. – 2012-10-29
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0Not in the definition of an ideal, but in the definition of a maximal ideal yes. – 2012-10-29
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0OK, I see, thank you very much – 2012-10-29