Universality of w-module analogues in commutative algebra

Prove or disprove that for every result in classical commutative algebra there exists a corresponding analogue within w-module theory as developed via star-operations (specifically the w-operation).

Background

The paper studies module-theoretic characterizations of Prüfer v-multiplication domains (PvMDs) in the w-operation framework and notes connections to longstanding questions proposed by Geroldinger–Kim–Loper. Within this context, the authors quote an explicit open problem from that work asking whether every classical result in commutative algebra admits a corresponding analogue in w-module theory.

As partial motivation and progress, the paper provides w-analogues of classical torsion and splitting criteria (e.g., w-purity of torsion submodules and w-splitting of canonical sequences for finitely generated or w-finitely generated modules in PvMDs), thereby addressing specific instances of the broader question but not settling the universal claim.