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 $4.$