Characterization of reflexive ideals in commutative rings

Establish a characterization of reflexive ideals in commutative rings, extending the numerical-semigroup characterization developed for ideals of numerical semigroups.

Background

The paper studies reflexive (equivalently, divisorial or bi-dual) ideals of numerical semigroups, where reflexivity is expressed through the numerical ideal operation as I = S − (S − I). For numerical semigroups, the authors characterize reflexive ideals as intersections of principal ideals and identify their irreducible members.

The authors explicitly place this result in the broader context of commutative ring theory and state that characterizing reflexive ideals for rings remains unresolved. Thus, the open problem is to obtain a corresponding general characterization in the ring-theoretic setting, beyond the numerical-semigroup case addressed in the paper.