A note on *_{𝑤}-Noetherian domains

Abstract

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