Reflexive rings and subrings of a product of Dedekind domains
Let \(R\) be a commutative Noetherian ring of Krull dimension 1, with zero nilradical. The full ring of quotients \(Q\) of \(R\) is then a direct product of fields, and the integral closure \(\overline R\) of \(R\) in \(Q\) is a direct product of Dedekind domains. Let \(A\) be an \(R\)-submodule of \(Q\) that contains a regular element, and let \({\mathcal C}_A\) be the category of modules that are \(\text{End}_R(A)\)-submodules of \(A^n\) for some \(n\). The author investigates the circumstances in which there is a Warfield duality on \({\mathcal C}_A\), that is, the functor \(\Hom_R(-,A)\) is a duality on \({\mathcal C}_A\). Given a pair of \(R\)-modules \(M,N\), say that \(M\) is \(N\)-reflexive if \(M\cong\Hom_R (\Hom_R (M,N), N)\). Say also that a fractional ideal \(I\) of \(R\) is canonical if \(\text{End}_R(I) \cap Q=R\) and every ideal of \(R\) is \(I\)-reflexive. Then the main characterizations are that Warfield duality holds for \({\mathcal C}_A\) iff any submodule of \(A\) is \(A\)-reflexive iff \(A_P\) is a canonical ideal of \(R_P\) for any maximal ideal \(P\) of \(R\). The author gives an example of a subring of an algebraic number field which is local and reflexive, but whose maximal ideal needs more than two generators, and an example of a non-reflexive ring having a canonical ideal.
scientific article; zbMATH DE number 1308353
scientific article