A note on *_w-noetherian domains

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-Noetheriandomains.