Kronecker function rings and power series rings

Abstract

Let D be an integral domain with quotient field K , X be an indeterminate over D , D [ [ X ] ] be the power series ring over D , and c ( f ) be the ideal of D generated by the coefficients of f ∈ D [ [ X ] ] . We will say that a star operation ∗ on D is a c-star operation if (i) c ( f g ) ∗ = ( c ( f ) c ( g ) ) ∗ for all 0 ≠ f , g ∈ D [ [ X ] ] and (ii) ( A B ) ∗ ⊆ ( A C ) ∗ implies B ∗ ⊆ C ∗ for all nonzero fractional ideals A , B , C of D . Assume that D admits a c-star operation ∗ , and let Kr ( ( D , ∗ ) ) = { f ∕ g ∣ f , g ∈ D [ [ X ] ] , g ≠ 0 , and c ( f ) ⊆ c ( g ) ∗ } . Among other things, we show that Kr ( ( D , ∗ ) ) is a Bezout domain, D is completely integrally closed, the v -operation on D is a c-star operation, and Kr ( ( D , v ) ) is a completely integrally closed Bezout domain. We also show that if V is a rank-one valuation domain, then the v -operation on V is a c-star operation, Kr ( ( V , v ) ) is a rank-one valuation domain, and Kr ( ( V , v ) ) is a DVR if and only if V is a DVR. Using this result, we show that if D is a generalized Krull domain, then Kr ( ( D , v ) ) is a one-dimensional generalized Krull domain.