Let $R$ be a principal ideal domain which is not a field, and let $M$ be a maximal ideal of the polynomial ring $R[X_1,\dots,X_n]$. If $n=1$ it is very easy to see that $M \cap R \neq 0$. Is this also true for $n>1$?
Contraction of maximal ideals in polynomial rings over PIDs
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ring-theory
commutative-algebra
principal-ideal-domains
maximal-and-prime-ideals
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4Dear Jose, Your "very easy to see" claim is wrong. E.g. if $R = \mathbb Z_p$ and $M$ is the ideal $(p x - 1)$ in $\mathbb Z_p[x]$, then $M$ is maximal, but has trivial intersection with $\mathbb Z_p$. Regards, – 2011-08-01