The Principal Ideal Theorem and Systems of Parameters

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.