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PreprintCC BY 4.0arXiv

The zero divisor and augmentation conjectures for medial quandle rings

  • Luc Ta
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Abstract

For quandle rings, Bardakov--Passi--Singh and Elhamdadi--Nunez--Singh respectively formulated an analogue of Kaplansky's zero divisor conjecture and a weakened analogue of Kaplansky's idempotent conjecture involving the augmentation map. In this note, we verify these conjectures for medial quandles. The latter holds for a larger class of quandles; we characterize integral domains over which the statement holds for this class. We also characterize integral domains of characteristic 0 over which the statement holds for a given finite Alexander quandle.

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Cite

BibTeXRIS
BibTeX
@misc{ta2026zero,
  title = {{The zero divisor and augmentation conjectures for medial quandle rings}},
  author = {Luc Ta},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.20306},
  eprint = {2610.20306},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.20306},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The zero divisor and augmentation conjectures for medial quandle rings
AU  - Luc Ta
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.20306
UR  - https://arxiv.org/abs/2610.20306
AB  - For quandle rings, Bardakov--Passi--Singh and Elhamdadi--Nunez--Singh respectively formulated an analogue of Kaplansky's zero divisor conjecture and a weakened analogue of Kaplansky's idempotent conjecture involving the augmentation map. In this note, we verify these conjectures for medial quandles. The latter holds for a larger class of quandles; we characterize integral domains over which the statement holds for this class. We also characterize integral domains of characteristic 0 over which the statement holds for a given finite Alexander quandle.
ER  -