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PreprintCC BY 4.0arXiv

A further study of quandles and quandle rings

  • Gregory Churchill
  • Indu Rasika Churchill
  • Neranga Fernando
  • Bhitali Kousik
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Abstract

We investigate core quandles and idempotents in quandle rings of core quandles. We answer several questions on the rank of core quandles and nontrivial idempotents in quandle rings. We also present solutions to two questions raised in a recent paper about non-trivial idempotents in quandle rings ℤ[R5] and ℤ[C5], where ℤ, R5 and C5 are the ring of integers, the dihedral quandle of order 5, and the commutative quandle of order 5, respectively. We then study units in extended quandle rings of a trivial quandle and the Joyce quandle, and nilpotent elements in extended quandles rings of a trivial quandle, where the ground ring is an integral domain. As a consequence, we show that the quandle ring and the extended quandle ring of a trivial quandle are not nil clean rings. We also explore prime rings and semi-prime rings among quandle rings. We introduce zero-divisor graphs of quandle rings and find an intriguing mirror symmetry among the in-degree and out-degree of the vertices. Moreover, we find a bivariate polynomial in the ring (ℤ2n+1[Q])[X,Y] that determines the commutative quandle of order 2n+1, where 2n+1 is prime.

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Cite

BibTeXRIS
BibTeX
@misc{churchill2026further,
  title = {{A further study of quandles and quandle rings}},
  author = {Gregory Churchill and Indu Rasika Churchill and Neranga Fernando and Bhitali Kousik},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.09149},
  eprint = {2608.09149},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.09149},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A further study of quandles and quandle rings
AU  - Gregory Churchill
AU  - Indu Rasika Churchill
AU  - Neranga Fernando
AU  - Bhitali Kousik
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.09149
UR  - https://arxiv.org/abs/2608.09149
AB  - We investigate core quandles and idempotents in quandle rings of core quandles. We answer several questions on the rank of core quandles and nontrivial idempotents in quandle rings. We also present solutions to two questions raised in a recent paper about non-trivial idempotents in quandle rings ℤ[R_5] and ℤ[C_5], where ℤ, R_5 and C_5 are the ring of integers, the dihedral quandle of order 5, and the commutative quandle of order 5, respectively. We then study units in extended quandle rings of a trivial quandle and the Joyce quandle, and nilpotent elements in extended quandles rings of a trivial quandle, where the ground ring is an integral domain. As a consequence, we show that the quandle ring and the extended quandle ring of a trivial quandle are not nil clean rings. We also explore prime rings and semi-prime rings among quandle rings. We introduce zero-divisor graphs of quandle rings and find an intriguing mirror symmetry among the in-degree and out-degree of the vertices. Moreover, we find a bivariate polynomial in the ring (ℤ_2n+1[Q])[X,Y] that determines the commutative quandle of order 2n+1, where 2n+1 is prime.
ER  -