Idempotents and Powers of Ideals in Quandle Rings

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.