Krull's Principal Ideal Theorem in non-Noetherian settings

TL;DR

This paper investigates the applicability of Krull's Principal Ideal Theorem in non-Noetherian rings, exploring conditions under which similar height bounds for ideals can be established beyond the classical Noetherian setting.

Contribution

It provides new insights and potential generalizations of Krull's theorem applicable to non-Noetherian rings, where the classical result does not hold.

Findings

Krull's theorem fails in non-Noetherian rings

Identifies conditions where height bounds can be extended

Proposes modified statements for non-Noetherian contexts

Abstract

Let $P$ be a finitely generated ideal of a commutative ring $R$. Krull's Principal Ideal Theorem states that if $R$ is Noetherian and $P$ is minimal over a principal ideal of $R$, then $P$ has height at most one. Straightforward examples show that this assertion fails if $R$ is not Noetherian. We consider what can be asserted in the non-Noetherian case in place of Krull's theorem.