In this paper we investigate idempotents in quandle rings and relate them with quandle coverings. We prove that integral quandle rings of non-trivial involutory coverings over nice base quandles have 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.

We show that quandle rings and their idempotents lead to proper enhancements of the well-known quandle coloring invariant of links in the 3-space. We give explicit examples to show that the new invariants are also stronger than the Hom quandle invariant when the coloring quandles are medial. We provide computer assisted computations of idempotents for all quandles of order less than six, and also determine quandles for which the set of all idempotents is itself a quandle. The data supports our conjecture about triviality of idempotents of integral quandle rings of finite latin quandles. We also determine Peirce spectra for complex quandle algebras of some small order quandles.

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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 nontrivial coverings over nice base quandles admit infinitely many nontrivial 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 nontrivial long knot admit nontrivial 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.