Idempotents in integral ring of dihedral quandle

TL;DR

This paper analyzes the structure of idempotents in the integral ring of dihedral quandles, providing complete solutions for specific cases and highlighting differences in automorphism groups.

Contribution

It offers a detailed description of idempotents in the integral ring of dihedral quandles, including a complete solution for n=5, and demonstrates a novel automorphism group property.

Findings

Complete description of idempotents for n=5

Identification of a connected quandle with non-isomorphic automorphism groups

Structural insights into dihedral quandle rings

Abstract

We provide the structure of some idempotents in the integral ring of a dihedral quandles, for odd case. The solution is complete in case $n=5$. This produces an example of a connected quandle $X$ such that $Aut (X)\not\cong Aut (\mathbb{Z}[X])$.