Integral closure, basically full closure, and duals of nonresidual closure operations

TL;DR

This paper develops a duality framework for closure operations on modules, extending previous dualities, and applies it to integral and basically full closures, providing new formulas and examples in algebra.

Contribution

It introduces a duality for module closure operations, generalizing prior results and applying it to integral and basically full closures with explicit formulas.

Findings

Derived dual formulas for integral and basically full closures.

Extended duality to include cores and hulls of modules.

Provided illustrative examples in a numerical semigroup ring.

Abstract

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.