Quandle rings

Abstract

In this paper, a theory of quandle rings is proposed for quandles analogous to the classical theory of group rings for groups, and interconnections between quandles and associated quandle rings are explored.

Ring theoretic aspects of quandles

“…In [3], power associativity of dihedral quandles was investigated and the question of determining the conditions under which the quandle ring R[X] is power associative was raised. In this section we give a complete solution to this question.…”

“…Recall that the fundamental quandles of knots are connected. Recently, there has been investigations of quandles from algebraic point of views and their relations to other algebraic structures such as Lie algebras [5,6], Leibniz algebras [15,16], Frobenius algebras and Yang-Baxter equation [7], Hopf algebras [2,6], transitive groups [28], quasigroups and Moufang loops [11], ring theory [3] etc. This article will add to this list since we introduce new concepts motivated by ring theory to the theory of quandles.…”

“…This article will add to this list since we introduce new concepts motivated by ring theory to the theory of quandles. We follow [3] and we associate to every quandle (X, ⊲) and an associative ring k with unity, a nonassociative ring k [X]. Precisely, let k[X] be the set of elements that are uniquely expressible in the form x∈X a x x, where x ∈ X and a x = 0 for almost all x.…”

“…Precisely, in studying self-distributivity maps in coalgebras, the authors of [5] In [3], the authors showed that the ring k[X] gives interesting information on the quandle X. In this article we investigate the basic properties of quandle rings and also solve some of the open problems stated in [3]. In particular, under the assumption that the inner automorphism group Inn(X) acts orbit 2-transitively on the quandle X and the ring k is a field of characteristic zero (or a certain semigroup H x acts 2-transitively on X) a complete description of right (or left) ideals is provided.…”

“…In Section 2, we recall the basics of quandles with examples. In Section 3, we investigate an open question raised in [3] concerning the power associativity of non-trivial quandles. Precisely, we prove that quandle rings are never power associative when the quandle is non-trivial and char(k) = 2, 3.…”