Levels of cancellation for monoids and modules

TL;DR

This paper investigates the levels of cancellativity in commutative monoids through stable rank values, analyzing their behavior, especially in refinement monoids and monoids derived from module classes, revealing structural properties.

Contribution

It characterizes stable rank behaviors in commutative monoids, especially refinement monoids, and explores their application to monoids formed from module isomorphism classes.

Findings

Stable rank values are determined for multiples in monoids.

Possible stable rank values in archimedean components are identified.

Stable rank behavior in module-based monoids is discussed.

Abstract

Levels of cancellativity in commutative monoids $M$, determined by stable rank values in $\mathbb{Z}_{> 0} \cup \{\infty\}$ for elements of $M$, are investigated. The behavior of the stable ranks of multiples $ka$, for $k \in \mathbb{Z}_{> 0}$ and $a \in M$, is determined. In the case of a refinement monoid $M$, the possible stable rank values in archimedean components of $M$ are pinned down. Finally, stable rank in monoids built from isomorphism or other equivalence classes of modules over a ring is discussed.