On the arithmetic of polynomial ideals

TL;DR

This paper explores the factorization properties of the monoid of ideals in multivariate polynomial rings, extending recent theories to identify new atomic structures and analyze their arithmetic characteristics.

Contribution

It introduces new methods for constructing atoms in ideal monoids and analyzes their arithmetic, especially for monomial ideals, advancing the understanding of ideal factorizations.

Findings

Constructed new families of atoms in ideal monoids.

Analyzed arithmetic properties of monomial ideals.

Computed sets of lengths for specific classes of ideals.

Abstract

This paper investigates atomic factorizations in the monoid $\mathcal I(R)$ of nonzero ideals of a multivariate polynomial ring $R$, under ideal multiplication. Building on recent advances in factorization theory for unit-cancellative monoids, we extend techniques from the paper [Geroldinger and Khadam, Ark. Mat. 60 (2022), 67-106] to construct new families of atoms in $\mathcal I(R)$, leading to a deeper understanding of its arithmetic. We further analyze the submonoid $\mathcal M\rm{on}(R)$ of monomial ideals, deriving arithmetic properties and computing sets of lengths for specific classes of ideals. The results advance the extensive study of ideal monoids within a classical algebraic framework.