Power series over Noetherian domains, Nagata rings, and Kronecker function rings

Let D be a Noetherian domain, ⁎ be a star operation on D, X be an indeterminate over D  , D[[X]] be the power series ring over D  , c(f)c(f) be the ideal of D   generated by the coefficients of f∈D[[X]], and N⁎={f∈D[[X]]|c(f)⁎=D}. Moreover, if ⁎ is e.a.b.  , then we let Kr((D,⁎))={fg|f,g∈D[[X]],0≠g, and c(f)⊆c(g)⁎}. In this paper, we show that N⁎N⁎ is a saturated multiplicative set and Kr((D,⁎)) is a Bezout domain. We then study some ring-theoretic properties of D[[X]]N⁎ and Kr((D,⁎)). For example, we prove that every invertible ideal of D[[X]]N⁎ is principal; dim(Kr((D,b)))=dimv(D); and if V is a valuation overring of D  , then Vˆ={fg|f,g∈D[[X]],g≠0, and c(f)V⊆c(g)V} is a valuation overring of D[[X]]; and Kr((D,⁎))=⋂{Vˆ|V is a ⁎-valuation overring of D}.