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

rle3791
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1 Answers1

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  1. Show that by localizing and factoring $A$ you can assume that $q_1 \cap A = 0$ and $q_3 \cap A$ is maximal (how?!).
  2. Assume $q_1 \cap A = q_3 \cap A$, then $A$ is a field (why?!).
  3. 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).
Jim
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