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.