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].

The important addition is Example 5.9 giving a negative answer to question 7.4 in [3] in characteristic zero. Theorem 3.3 (from the previous version) was corrected and it's now Theorem 3.4

References in corpus (9)

• Leibniz algebras, Lie racks, and digroups
• Automorphism groups of Quandles
• Two-point homogeneous quandles with prime cardinality
• Doubly transitive groups and cyclic quandles
• Quandle rings
• Causality from dynamical symmetry: an example from local scale-invariance
• Distributivity in Quandles and Quasigroups
• Connected Quandles with Order Equal to Twice an Odd Prime
• Cohomology of Categorical Self-Distributivity

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• A Survey of Racks and Quandles: Some recent developments
• Counterexample to a conjecture about dihedral quandle
• The matched product of set-theoretical solutions associated with shelves
• Legendrian Rack Invariants of Legendrian Knots