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If $S = \langle2\rangle$ is the ideal generated by $2$ in $\mathbb{Z}$, what does $S[x]$ represent?

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    $\begingroup$ Presumably the ideal of polynomials with even coefficients. $\endgroup$ – Potato Apr 18 '14 at 6:13
  • $\begingroup$ but in my problem, it says to prove that S[x] is not maximal. isn't the ideal you just said maximal? $\endgroup$ – Paul Malinowski Apr 18 '14 at 6:14
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    $\begingroup$ I don't think so. Consider the ideal $I$ of all polynomials with even constant term and possibility odd coefficients on the $x^n$ terms. Then $S\subset I$. $\endgroup$ – Potato Apr 18 '14 at 6:16
  • $\begingroup$ Ahh! of course! thank you $\endgroup$ – Paul Malinowski Apr 18 '14 at 6:23
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$S[x]$ is the ideal of polynomials with even coefficients. Generally if $I \subset R$ is an ideal and $M$ is a module, the $IM$ is the submodule of $M$ generated by elements of the form $im$, where $i \in I$ and $m \in M$. Sometimes you can use shortcuts, for example $I R[x] =: I[x]$.

It's not maximal in $\mathbb{Z}[x]$: the quotient $\mathbb{Z}[x] / S[x]$ is equal to $(\mathbb{Z}/2\mathbb{Z})[x]$, which is not a field. Another way of seeing it is to notice that $S[x]$ is included in the ideal generated by $2$ and $x$ -- in other words the ideal of polynomials with even constant term.

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