  In this paper, we investigate idempotents in quandle rings and relate them
with quandle coverings. We prove that integral quandle rings of quandles of
finite type that are non-trivial coverings over nice base quandles admit
infinitely many non-trivial idempotents, and give their complete description.
We show that the set of all these idempotents forms a quandle in itself. As an
application, we deduce that the quandle ring of the knot quandle of a
non-trivial long knot admit non-trivial idempotents. We consider free products
of quandles and prove that integral quandle rings of free quandles have only
trivial idempotents, giving an infinite family of quandles with this property.
We also give a description of idempotents in quandle rings of unions and
certain twisted unions of quandles.

Rings and Algebras;Group Theory;Geometric Topology;Quantum Algebra