Ring Theoretic Aspects of Quandles

TL;DR

This paper explores the algebraic structure of quandle rings associated with quandles, providing classifications of ideals under certain symmetry conditions, and presents examples that distinguish quandle isomorphism from ring isomorphism.

Contribution

It offers a detailed analysis of the properties of quandle rings, including ideal classifications and conditions for ring isomorphisms implying quandle isomorphisms, addressing open problems in the field.

Findings

Complete description of right ideals for dihedral quandles

Conditions under which quandle rings determine quandle isomorphism

Examples of non-isomorphic quandles with isomorphic rings

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].