The zero divisor and augmentation conjectures for medial quandle rings

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.