A text I am following uses of the following (probably basic) commutative algebraic lemma, omitting its proof.

Lemma: Let $n\in\mathbb{N}_{>0}$, and let $P_0\subsetneq P_1\subsetneq\cdots\subsetneq P_n$ be a chain of prime ideals in a Noetherian ring $R$. Moreover, let $a\in P_n$. Then there is a chain of prime ideals $P_0'\subsetneq P_1'\subsetneq\cdots\subsetneq P_{n-1}'\subsetneq P_n$ (i. e., in the given chain we may change all prime ideals except the last one) so that $a\in P_1'$.

Given its place in the text, there should be a proof of this using nothing more advanced than Krull's Principal Ideal Theorem. However I have not been able to find such a proof.

Any help?


A proof is given in Bourbaki, Algèbre commutative, VIII.3.1 Lemme 1.

  • $\begingroup$ For clarity: page AC VIII.25. $\endgroup$ – Alex M. Jun 30 '19 at 16:06

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