-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 KxJ 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.