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).

Keywords: *w-Noetherian domain, *w