∗-Noetherian domains and the ring D[X]N∗

Let D be an integral domain with quotient field K, ∗ a star-operation on D, X a nonempty set of indeterminates over D, and N∗={f∈D[X]|∗(Af)=D}. For a nonzero fractional ideal I of D, let I∗w={x∈K|xJ⊆I for J a nonzero finitely generated ideal of D with J∗=D}; then ∗w is a finite character star-operation on D. We prove that D is a ∗w-Noetherian domain if and only if each prime ∗w-ideal of D is of finite type, if and only if D[X]N∗ is a Noetherian domain. We also study the ∗-global transform, ∗-linked overrings, and the ∗w-integral closure of a ∗w-Noetherian domain.