  We develop a duality for operations on nested pairs of modules that
generalizes the duality between absolute interior operations and residual
closure operations from [ER21], extending our previous results to the expanded
context. We apply this duality in particular to integral and basically full
closures and their respective cores to obtain integral and basically empty
interiors and their respective hulls. We also dualize some of the known
formulas for the core of an ideal to obtain formulas for the hull of a
submodule of the injective hull of the residue field. The article concludes
with illustrative examples in a numerical semigroup ring.
