On the Krull Intersection Theorem in Function Algebras

TL;DR

This paper explores the Krull Intersection Theorem within various function algebras, identifying conditions under which the intersection of powers of ideals is trivial or non-trivial, extending classical algebraic results.

Contribution

It investigates the validity of the Krull Intersection Theorem in function algebras and characterizes ideals with non-trivial intersections.

Findings

Identifies function algebras where the theorem holds or fails.

Provides criteria for when the intersection of ideal powers is zero.

Examples of ideals with non-zero Krull intersections.

Abstract

A version of the Krull Intersection Theorem states that for Noetherian domains, the Krull intersection $ki(I)$ of every proper ideal $I$ is trivial; that is $$ ki(I):=\displaystyle\bigcap_{n=1}^\infty I^n = \{0\}. $$ We investigate the validity of this result for various function algebras $R$, present ideals $I$ of $R$ for which $ ki(I)\neq \{0\}$, and give conditions on $I$ so that $ki(I)=\{0\}$.