Question: To prove that the Ring of Complex Entire functions is neither Artinian nor noetherian.
Proof: Clearly $R$ is not Artinian because it is a commutative integral domain which is not a field, and $R$ is not noetherian because its not a factorisation domain.
Is there a proof of this theorem using the Ascending Chain / Descending Chain condition for Artinian / Noetherian rings?