相关论文: Idempotents in integral ring of dihedral quandle

This article addresses two central problems in the theory of quandle rings. First, motivated by Conjecture 3.10 in Internat. J. Math. 34 (2023), no. 3, Paper No. 2350011: for a semi-latin quandle $X$, every nonzero idempotent in the…

We associate to every quandle $X$ and an associative ring with unity $\mathbf{k}$, a nonassociative ring $\mathbf{k}[X]$ following [3]. The basic properties of such rings are investigated. In particular, under the assumption that the inner…

The paper develops further the theory of quandle rings which was introduced by the authors in a recent work. Orderability of quandles is defined and many interesting examples of orderable quandles are given. It is proved that quandle rings…

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…

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 $\Hom$…

In the present work, a procedure for determining idempotents of a commutative ring having a sequence of ideals with certain properties is presented. As an application of this procedure, idempotent elements of various commutative rings are…

Idempotents yield much insight in the structure of finite semigroups and semirings. In this article, we obtain some results on (multiplicatively) idempotents of the endomorphism semiring of a finite chain. We prove that the set of all…

We prove that the displacement group of the dihedral quandle with n elements is isomorphic to the group generated by rotations of the n/2-gon when n is even and the n-gon when n is odd. We additionally show that any quandle with at least…

Let $K = \mathbb R$ or $\mathbb C$ and $\mathcal R_n$ be the dihedral quandle of order $n$: In this article, we give decomposition of the quandle ring $K[\mathcal R_n]$ into indecomposable right $K[\mathcal R_n]$-modules for all even $n \in…

We study a ring containing a complete set of orthogonal idempotents as a generalized matrix ring via its Peirce decomposition. We focus on the case where some of the underlying bimodule homomorphisms are zero. Upper and lower triangular…

Commutative semirings with divisible additive semigroup are studied. We show that an additively divisible commutative semiring is idempotent, provided that it is finitely generated and torsion. In case that a one-generated additively…

We systematically study those rings whose non-units are a sum of an idempotent and a nilpotent. Some crucial characteristic properties are completely described as well as some structural results for this class of rings are obtained. This…

We show that an idempotent lies in the center if it commutes with the other idempotents in the ring. Next, we introduce a partition of the set of idempotents and show that the automorphisms of the ring act transitively on each equivalence…

We prove that a semiring multiplicatively generated by its idempotents is commutative and Boolean, if every idempotent in the semiring has an orthogonal complement. We prove that a semiring additively generated by its idempotents is…

We present new characterizations of the rings in which every element is the sum of two idempotents and a nilpotent that commute, and the rings in which every element is the sum of two tripotents and a nilpotent that commute. We prove that…

We prove that the automorphism group of the dihedral quandle with n elements is isomorphic to the affine group of the integers mod n, and also obtain the inner automorphism group of this quandle. In [9], automorphism groups of quandles (up…

In this note we answer the question raised by Han et al. in J. Korean Math. Soc (2014) whether an idempotent isomorphic to a semicentral idempotent is itself semicentral. We show that rings with this property are precisely the…

We investigate properties of the group inverse in rings with unit related to products and differences of idempotents, and thus we extend some results from \cite{DENG} to more general settings. We show that most part of \cite{DENG} is easily…

Quandles are self-distributive, right-invertible, idempotent algebras. A group with conjugation for binary operation is an example of a quandle. Given a quandle $(Q, \ast)$ and a positive integer $n$, define $a\ast_n b = (\cdots (a\ast…

We are intereseted in quandles and their enveloping groups. Various results are proven. We show that a quandle $Q$ and its image in the enveloping group $G(Q)$ have isomorphic enveloping groups. The image quandle is injective. For $Q$ a…