The Principal Ideal Theorem and Systems of Parameters

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


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.


Prime Ideal Local Ring Maximal Ideal Polynomial Ring Hilbert Series 
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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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