Decomposition of quandle rings of dihedral quandles

TL;DR

This paper studies the algebraic structure of quandle rings associated with dihedral quandles, providing a detailed decomposition into indecomposable modules specifically for even orders, and clarifying conditions under which previous decompositions hold.

Contribution

It offers a complete decomposition of dihedral quandle rings into indecomposable modules for all even n, refining earlier results by identifying when certain decompositions are valid.

Findings

Decomposition of $K[ ext{dihedral quandle}]$ into indecomposable modules for even n.

Previous decompositions are valid only when n is not divisible by 4.

Provides explicit module decomposition for all even n.

Abstract

Let $K = \mathbb R$ or $\mathbb C$ and $\mathcal R_n$ be the dihedral quandle of order $n$: In this article, we give decomposition of the quandle ring $K[\mathcal R_n]$ into indecomposable right $K[\mathcal R_n]$-modules for all even $n \in \mathbb N$. It follows that the decomposition of $K[\mathcal R_n]$ given in [EFT19, Prop. 4.18(2)] is valid only in the case when $n$ is not divisible by $4.$