Divisor sequences of atoms in Krull monoids

TL;DR

This paper explores which integer sequences can represent the divisor sequences of atoms in Krull monoids, providing insights into non-unique module decompositions where classical uniqueness properties fail.

Contribution

It characterizes realizable divisor sequences of atoms in Krull monoids and connects these sequences to module decomposition problems in algebra.

Findings

Identifies possible divisor sequences for atoms in Krull monoids

Links divisor sequences to non-unique module decompositions

Provides a framework for studying failure of Krull-Remak-Schmidt property

Abstract

The divisor sequence of an irreducible element (\textit{atom}) $a$ of a reduced monoid $H$ is the sequence $(s_n)_{n\in \mathbb{N}}$ where, for each positive integer $n$, $s_n$ denotes the number of distinct irreducible divisors of $a^n$. In this work we investigate which sequences of positive integers can be realized as divisor sequences of irreducible elements in Krull monoids. In particular, this gives a means for studying non-unique direct-sum decompositions of modules over local Noetherian rings for which the Krull-Remak-Schmidt property fails.