Factorization in the Monoid of Integrally Closed Ideals

TL;DR

This paper studies the algebraic structure of integrally closed ideals in Noetherian rings, especially polynomial rings, revealing their monoid properties, factorization, and connections to polytope groups, with implications for explicit ideal factorization.

Contribution

It introduces a new monoid structure on integrally closed ideals, proves their atomicity and torsion-free nature, and establishes a surjective homomorphism linking these ideals to the integral polytope group.

Findings

The monoid of integrally closed ideals is cancellative, torsion-free, and atomic.

Every such ideal containing a nonzerodivisor can be factored into irreducibles.

A surjective homomorphism connects the Grothendieck group of monomial ideals to the integral polytope group.

Abstract

Given a Noetherian ring $A$, the collection of all integrally closed ideals in $A$ which contain a nonzerodivisor, denoted $ic(A)$, forms a cancellative monoid under the operation $I*J=\overline{IJ}$, the integral closure of the product. The monoid is torsion-free and atomic -- every integrally closed ideal in $A$ containing a nonzerodivisor can be factored in this $*$-product into $*$-irreducible integrally closed ideals. Restricting to the case where $A$ is a polynomial ring and the ideals in question are monomial, we show that there is a surjective homomorphism from the Integral Polytope Group onto the Grothendieck group of integrally closed monomial ideals under translation invariance of their Newton Polyhedra. Notably, the Integral Polytope Group, the Grothendieck group of polytopes with integer vertices under Minkowski addition and translation invariance, has an explicit basis, allowing for explicit factoring in the monoid.