Finite Modules over $\Bbb Z[t,t^{-1}]$

TL;DR

This paper classifies finite modules over the Laurent polynomial ring over integers for orders up to p^4, providing a comprehensive understanding of their structure and related Alexander quandles.

Contribution

It offers a complete classification of finite $Z[t,t^{-1}]$-modules of order p^n for n ≤ 4, a new result in module theory and knot invariants.

Findings

Classification of modules of order p^n for n ≤ 4

Enumeration of Alexander quandles of order p^n for n ≤ 4

Structural insights into modules over Laurent polynomial rings

Abstract

Let $\Lambda=\Bbb Z[t,t^{-1}]$ be the ring of Laurent polynomials over $\Bbb Z$. We classify all $\Lambda$-modules $M$ with $|M|=p^n$, where $p$ is a primes and $n\le 4$. Consequently, we have a classification of Alexander quandles of order $p^n$ for $n\le 4$.