Decomposition of quandle rings of dihedral quandles

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 $4.$