Reflexive D-rings
A domain is called reflexive if every torsionless module of finite rank is reflexive, and D-rings R are domains with the following property: Every torsion-free R-module of finite rank is a direct sum of rank 1 modules. Matlis characterized several classes of D-rings, among them the integrally closed and the Noetherian ones. A general characterization of D-rings is lacking however. It follows from Matlis' results that every quasi-local D-ring is an intersection of quasi-local reflexive D-rings contained in its integral closure. This leads the author to study quasi- local reflexive D-rings. In the following Q denotes the field of fractions of R. We quote some of the results obtained: (1) A quasi-local D-ring is reflexive if and only if each ideal is reflexive. (2) Let R be quasi- local, V its integral closure, and I the conductor; then R is reflexive if and only if V and R/I are maximal valuation rings, Q is a linearly compact R-module, and all ideals of R are reflexive. (3) With the notations of (2), R is a reflexive D-ring if and only if V and R/I are maximal valuations rings and Q/R is the injective envelope of the residue class field of R.
scientific article; zbMATH DE number 4125524
scientific article