On the Krull Intersection Theorem in Function Algebras

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 $ki(I)=\{0\}$.