Idempotents and Powers of Ideals in Quandle Rings

TL;DR

This paper investigates the structure of idempotents and powers of ideals in quandle rings, proving new results about trivial idempotents in certain cases and extending computations to dihedral and commutative quandles.

Contribution

It proves that quandle rings of Core($\

Findings

Only trivial idempotents in quandle rings of Core($\

powers of augmentation ideals computed for dihedral and commutative quandles

Automorphism groups of integral quandle rings of 2-almost latin quandles determined

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.