Kaur, Dilpreet

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

Let (Formula presented.) be field of complex numbers and X be a connected quandle of prime order. We study the regular representation of X by describing the quandle ring (Formula presented.) as direct sum of right simple ideals. We provide description of tensor product of connected quandles of prime order. We further discuss multiplicity freeness of quandle ring decomposition for connected quandles of order (Formula presented.) and prove that (Formula presented.) decomposes multiplicity free for affine connected quandle X.