On the Intersection Algebra of Principal Ideals

TL;DR

This paper investigates the algebraic structure of intersections of principal ideals in a UFD, introduces fan algebras, and provides an algorithm for generating these intersection algebras, with implementation in Macaulay2.

Contribution

It presents a new algorithm for generating intersection algebras of principal ideals and introduces fan algebras, expanding computational tools in commutative algebra.

Findings

Algorithm successfully generates intersection algebras over UFDs.

Implementation available in Macaulay2 for polynomial rings.

Introduction of fan algebras as a new class of algebras.

Abstract

We study the finite generation of the intersection algebra of two principal ideals I and J in a unique factorization domain R. We provide an algorithm that produces a list of generators of this algebra over R. In the special case that R is a polynomial ring, this algorithm has been implemented in the commutative algebra software system Macaulay2. A new class of algebras, called fan algebras, is introduced.