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

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## Copyright information

© Springer-Verlag New York, Inc. 1995