Group-ring methods for Alexander quandle rings
Abstract
For an Alexander quandle on an abelian group G with automorphism ϕ, we identify its augmentation powers and nonunital commutator subalgebra with explicit ideals in the ordinary group ring over any commutative unital coefficient ring. Each left-normed augmentation power is also the span of all products with the same number of factors, independently of parenthesization. Over ℤ, we determine every successive augmentation quotient for Alexander quandles on nontrivial finite cyclic groups. For an even dihedral quandle of order m ≥ 4, every quotient in degree r ≥ 2 is (ℤ/(m/2)ℤ)2; this proves the proposed replacement for the disproved order-m prediction. For the integral quandle ring of Core(ℤ), we describe all augmentation powers, determine the nonunital commutator subalgebra, and prove that the difference of the basis elements indexed by 1 and 0 does not lie in the square of the augmentation ideal. When (1−ϕ)G is a finite p-group, the integral filtration is separated, so every nonzero integral idempotent has augmentation one. Fourier analysis shows that finite odd-order commutative Alexander quandles have only zero and the basis elements as integral idempotents. Combined with the midpoint description of medial commutative quandles, this gives the same classification for every finite medial commutative quandle. Finally, the complex quandle algebra of the core of any nontrivial finite abelian group of odd order has right Peirce spectrum {0,1,−1}, proving the conjectured odd-order dihedral spectrum.
Related
- Derivations of Lie p-rings with applications on ring extensions and computational methods
- A note on idempotents in quandle rings
- The zero divisor and augmentation conjectures for medial quandle rings
- A further study of quandles and quandle rings
- The Hilbert Basis Theorem for Non-Associative Skew Power Series Rings and Non-Associative Skew Laurent Series Rings over a Non-Associative Coefficient Ring
BibTeX
@misc{zhang2026groupring,
title = {{Group-ring methods for Alexander quandle rings}},
author = {Zhi-Lin Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08204},
eprint = {2610.08204},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08204},
}
RIS
TY - UNPB
TI - Group-ring methods for Alexander quandle rings
AU - Zhi-Lin Zhang
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08204
UR - https://arxiv.org/abs/2610.08204
AB - For an Alexander quandle on an abelian group G with automorphism ϕ, we identify its augmentation powers and nonunital commutator subalgebra with explicit ideals in the ordinary group ring over any commutative unital coefficient ring. Each left-normed augmentation power is also the span of all products with the same number of factors, independently of parenthesization. Over ℤ, we determine every successive augmentation quotient for Alexander quandles on nontrivial finite cyclic groups. For an even dihedral quandle of order m ≥ 4, every quotient in degree r ≥ 2 is (ℤ/(m/2)ℤ)^2; this proves the proposed replacement for the disproved order-m prediction. For the integral quandle ring of Core(ℤ), we describe all augmentation powers, determine the nonunital commutator subalgebra, and prove that the difference of the basis elements indexed by 1 and 0 does not lie in the square of the augmentation ideal. When (1−ϕ)G is a finite p-group, the integral filtration is separated, so every nonzero integral idempotent has augmentation one. Fourier analysis shows that finite odd-order commutative Alexander quandles have only zero and the basis elements as integral idempotents. Combined with the midpoint description of medial commutative quandles, this gives the same classification for every finite medial commutative quandle. Finally, the complex quandle algebra of the core of any nontrivial finite abelian group of odd order has right Peirce spectrum {0,1,−1}, proving the conjectured odd-order dihedral spectrum.
ER -