Commutative Algebra pp 231-246 | Cite as

# The Principal Ideal Theorem and Systems of Parameters

Chapter

## 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, *Q* ⫋ *P* ⫋ (*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 *y* ∈ *P*, then *y* = *ax* for some *a*, and since *x* ∉ *P* it follows that *a* ∈ *P*; 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