Idempotents and Powers of Ideals in Quandle Rings

TL;DR

This work advances quandle-ring theory by characterizing idempotents and augmentation-ideal structures across several quandle families. It proves that, over an integral domain with unity, the quandle ring of the Core quandle of $\mathbb{Z}$ has only trivial idempotents, and it extends augmentation-ideal power calculations to dihedral and commutative quandles while detailing idempotent behavior in $2$-almost latin quandles and their automorphism groups. The results provide explicit bases for augmentation-ideal powers in key cases (e.g., dihedral and commutative quandles) and connect quandle properties to the structure of their integral quandle rings. Overall, the paper deepens understanding of idempotents, augmentation filtrations, and automorphisms in quandle-ring contexts and maps out several avenues for future work.

Abstract

This article addresses two central problems in the theory of quandle rings. First, motivated by Conjecture 3.10 in Internat. J. Math. 34 (2023), no. 3, Paper No. 2350011: for a semi-latin quandle $X$, every nonzero idempotent in the integral quandle ring $\mathbb{Z}[X]$ necessarily corresponds to an element of $X$, we investigate idempotents in quandle rings of semi-latin quandles. Precisely, we prove that if the ground ring is an integral domain with unity, then the quandle ring of Core($\mathbb{Z}$) admits only trivial idempotents. Second, powers of augmentation ideals in quandle rings have only been computed in few cases previously. We extend the computations to include dihedral quandles and commutative quandles. Finally, we examine idempotents in quandle rings of $2$-almost latin quandles and apply these results to compute the automorphism groups of their integral quandle rings.

Theorems & Definitions (34)

• Definition 2.1
• Conjecture 2.2
• Proposition 2.3
• Proposition 2.4
• Theorem 3.1
• proof
• Proposition 3.5
• proof
• Lemma 4.1
• Proposition 4.2