On the arithmetic of polynomial ideals

TL;DR

The paper studies atomic factorizations in the unit-cancellative monoid ${\mathcal I}(R)$ of nonzero ideals of a multivariate polynomial ring $R$, and analyzes the submonoid ${\rm Mon}(R)$ of monomial ideals. Building on unit-cancellative factorization theory, it constructs new atoms in ${\mathcal I}(R)$ via sum-free subsets of $\mathbb N$ and introduces a novel atom family $\tilde{\mathfrak{b}}_r$ in ${\rm Mon}(R)$, establishing precise sets of lengths for key families such as $I_{C_n}$ and $\langle X^m,Y^n\rangle$. The monoid ${\rm Mon}(R)$ is shown to be a BF-monoid and fully elastic but not transfer Krull, with detailed length behavior ${\mathsf L}_{\mathcal{M}{\rm on}}(\mathfrak a_k)=\llbracket 2,k\rrbracket$ for $k\ge 2$ and ${\mathsf L}_{\mathcal{M}{\rm on}}(I_{C_n})=\llbracket 2,n+1\rrbracket$; these results illuminate the arithmetic of ideal monoids within a classical algebraic framework and connect to the power-monoid of $\mathbb N$ via explicit monoid maps.

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.

Theorems & Definitions (47)

• Lemma 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Lemma 2.5: Lemma 5.6 of Ge-Kh21
• Remark 2.6
• Lemma 3.1
• proof
• Lemma 3.2
• proof