Divisor sequences of atoms in Krull monoids

Abstract

The divisor sequence of an irreducible element (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 $a^n$. 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 nonunique direct-sum decompositions of modules over local Noetherian rings for which the Krull–Remak–Schmidt property fails.