I have a question regarding an exercise I found in Eisenbud's Commutative Algebra with a view towards Algebraic Geometry:

Exercise 17.6: Any ideal of a Noetherian ring generated by a regular sequence can be generated by a regular sequence in any order (i.e. permutations of the regular sequence still form a regular sequence).

In the section "Hints and Solutions for Selected Exercises", there are hints on how to solve the exercise, which break the proof down to 4 separate steps. Steps 1, 2 and 4 are manageable, but I am kind of stuck at step 3:

if $x_1,\ldots,x_r$ is a regular sequence $\Rightarrow \exists j \in (x_1,\ldots,x_{r-1})$ such that $x_r+j$ is a nonzerodivisor modulo every subset of $\{x_1,\ldots,x_{r-1}\}$.

The hint he stated ("use prime avoidance in the form given in Exercise 3.8b") is not that useful, since there is no Exercise 3.8b. I tried to find out what else he might have referred to but I couldn't find anything applicable, so I am stuck with the hint that some sort of prime avoidance is necessary.

One can write the set of zero divisors as a union of prime ideals, the radicals of the primary ideals which decompose $(0)$, but I also fail to see how it might help.

I hope somebody can help me with this problem that has been bugging me for the past week.

Thanks in advance!

  • 3
    $\begingroup$ Maybe Exercise 3.8b was meant to be Exercise 3.19b? $\endgroup$
    – Andrew
    Jul 13, 2012 at 17:23
  • 3
    $\begingroup$ Exercise 3.19b solves Step 3 immediately. One can also look for the exercise 1.2.21 from Bruns & Herzog, Cohen-Macaulay Rings. $\endgroup$
    – user26857
    Jul 13, 2012 at 20:54


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