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The Principal Ideal Theorem and Systems of Parameters

  • David Eisenbud
Part of the Graduate Texts in Mathematics book series (GTM, volume 150)

Abstract

It is elementary that a principal prime ideal in a Noetherian ring can have codimension at most 1. A sharper statement is this: Any prime properly contained in a proper principal ideal has codimension 0. Proof: If on the contrary, QP ⫋ (x) in a ring R, with P and Q prime, then factoring out Q we can assume that Q = 0, and thus that R is a domain. If yP, then y = ax for some a, and since xP it follows that aP; thus P = xP. By Corollary 4.7, (1 - b)P = 0 for some b ∈ (x). Since R is a domain, we must have b = 1, so (x) is not proper, a contradiction.

Keywords

Prime Ideal Local Ring Maximal Ideal Polynomial Ring Hilbert Series 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York, Inc. 1995

Authors and Affiliations

  • David Eisenbud
    • 1
  1. 1.Department of MathematicsBrandeis UniversityWalthamUSA

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