Ring Theoretic Aspects of Quandles

Abstract

We associate to every quandle $X$ and an associative ring with unity $\mathbf{k}$, a nonassociative ring $\mathbf{k}[X]$ following [3]. The basic properties of such rings are investigated. In particular, under the assumption that the inner automorphism group $Inn(X)$ acts orbit $2$-transitively on $X$, a complete description of right (or left) ideals is provided. The complete description of right ideals for the dihedral quandles $R_n$ is given. It is also shown that if for two quandles $X$ and $Y$ the inner automorphism groups act $2$-transitively and $\mathbf{k}[X]$ is isomorphic to $\mathbf{k}[Y]$, then the quandles are of the same partition type. However, we provide examples when the quandle rings $\mathbf{k}[X]$ and $\mathbf{k}[Y]$ are isomorphic, but the quandles $X$ and $Y$ are not isomorphic. These examples answer some open problems in [3].