For a maximal ideal $n$ of a polynomial ring $ R [x] $ over a commutative ring $R$ with identity, are there conditions under which $m [x]\subset n$, for some maximal ideal $m$ of $R$?

Note: $m [x] $ is the ideal of $R [x] $, generated by $m $.

For example, if $R $ is a zero dimensional ring it is true.


This is equivalent to $m\subset n\cap R$ ; equivalently, it means that $n\cap R$ (which is a prime ideal of $R$) is maximal. This holds in particular if $R$ is a Jacobson ring, that is, every prime ideal is an intersection of maximal ideals; every finitely generated algebra over a field or over $\mathbb{Z}$ is a Jacobson ring. For all of this see for instance EGA I, §6.4.

  • $\begingroup$ @ abx: Did you mean $n\cap R$ or $n\cap R[x]$? $\endgroup$ – Mahdi Majidi-Zolbanin Dec 11 '15 at 14:09
  • $\begingroup$ Oups! That was $n\cap R$ of course, thanks. I edit. $\endgroup$ – abx Dec 11 '15 at 14:42

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