On the arithmetic of polynomial 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 $\mathcal{M}\mathrm{on}(\mathrm{R})$ of nonzero 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.

Revised version after peer review: several minor corrections, a substantial simplification of the proofs of Lemmas 3.8, 3.9 and 3.10, and the addition of Proposition 4.5