On ∗w-Finiteness Conditions.

Let D be an integral domain and let ∗ be a star-operation on D. In this article, we give new characterizations of ∗w-Noetherian domains and ∗w-principal ideal domains. More precisely, we show that D is a ∗w-Noetherian domain (resp., ∗w-principal ideal domain) if and only if every ∗w-countable type ideal of D is of ∗w-finite type (resp., principal). [ABSTRACT FROM AUTHOR]