Exercise about prime ideals in a polynomial ring Are considered prime ideals $q_{1}\subsetneqq q_{2}\subsetneqq q_{3} \subseteq A[X]$. Could you show that $q_{1}\cap A\neq q_{3}\cap A$ ?

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*Show that by localizing and factoring $A$ you can assume that $q_1 \cap A = 0$ and $q_3 \cap A$ is maximal (how?!).

*Assume $q_1 \cap A = q_3 \cap A$, then $A$ is a field (why?!).

*If $A$ is a field show that $q_1 \subsetneq q_2 \subsetneq q_3$ can't happen (use Krull dimension if you know what that is, if you don't use that $A[x]$ is a PID).

