Divisor sequences of atoms in Krull monoids

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.

Theorems & Definitions (29)

• Definition 2.1
• Example 2.2
• Remark 2.3
• Proposition 2.4: WW09, Proposition 2.7
• Proposition 2.5
• Remark 2.6
• Lemma 2.7
• proof
• Theorem 3.1
• proof