Strong Mori domains and the ring D[X]Nv

Let D be an integral domain with quotient field K, I a nonzero fractional ideal of D, X a nonempty set of indeterminates over D, and Nv={fset membership, variantD[X](Af)v=D}. In this paper, we show that ID[X]Nv∩K=Iw; Iw is of finite type if and only if ID[X]Nv is finitely generated; and D is an strong Mori domain (SM-domain) if and only if D[X]Nv is a Noetherian domain. Using these results, we give several Noetherian-like properties of SM-domains.