# maximal ideals of polynomial ring

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