Let D be an integral domain with quotient field K, * be a staroperation on D, and GV * (D) be the set of finitely generated ideals J of D such that J * = D. Then the map * w defined by I * w = {x ∈ K | Jx ⊆ I for some J ∈ GV * (D)} for all nonzero fractional ideals I of D is a finite character staroperation on D. In this paper, we study several properties of * w -Noetherian domains. In particular, we prove the Hilbert basis theorem for * w -Noetherian domains.