Zero-divisors and idempotents in quandle rings

幂等元与理想在格架环中的幂

Idempotents and Powers of Ideals in Quandle Rings

本文研究了半拉丁格架环中的幂等元性质，并扩展了扩增理想幂的计算，应用于自同构群的计算。

关于quandle与quandle环的进一步研究

A further study of quandles and quandle rings

本文针对quandle及quandle环，解答了其秩、非平凡幂等元等问题，证明平凡quandle的两类环非幂零clean环，还研究了其零因子图与相关多项式。

中顶quandle环的零因子与增广猜想

The zero divisor and augmentation conjectures for medial quandle rings

本文验证了中顶quandle的零因子与增广猜想，刻画了相关类quandle及有限Alexander quandle对应陈述成立的整环，推进了quandle环相关猜想的研究。

Determination of the biquaternion divisors of zero, including the idempotents and nilpotents

The biquaternion (complexified quaternion) algebra contains idempotents (elements whose square remains unchanged) and nilpotents (elements whose square vanishes). It also contains divisors of zero (elements with vanishing norm). The idempotents and nilpotents are subsets of the divisors of zero. These facts have been reported in the literature, but remain obscure through not being gathered together using modern notation and terminology. Explicit formulae for finding all the idempotents, nilpotents and divisors of zero appear not to be available in the literature, and we rectify this with the p…

Alexander quandle 环的群环方法

Group-ring methods for Alexander quandle rings

本文用群环方法研究 Alexander quandle 环，识别增广幂与交换子子代数，确定有限循环及二面体 quandle 的增广商，并分类幂等元与证明 Peirce 谱猜想。

无挠群分次环中的单位、零因子与幂等元

Units, zero-divisors and idempotents in rings graded by torsion-free groups

本文将无挠群群环的三大经典Kaplansky问题推广至无挠群分次环，限定研究无非平凡齐次零因子的分次环，解决了唯一积群分次环的情形，证明了扩展问题与经典问题层级一致，并得到分次环的不可分解性等性质，推广了经典群环猜想。

分级环中零因子、单位和幂等元的齐性

Homogeneity of zero-divisors, units and idempotents in a graded ring

本文受Kaplansky猜想启发，证明了分级环（尤其是monoid-rings）的若干重要结果，将群环 $K[G]$ 的Kaplansky零因子猜想推广至无挠群 $G$ 的 $G$-分级环情形。

Uniquely weakly nil-clean conditions on zero-divisors

An element in a ring $R$ is called uniquely weakly nil-clean if every element in $R$ can be uniquely written as a sum or a difference of a nilpotent and an idempotent in the sense of very idempotents. The structure of the ring in which every zero-divisor is uniquely weakly nil-clean is completely determined. We prove that every zero-divisor in a ring $R$ is uniquely weakly nil-clean if and only if $R$ is a D-ring, or $R$ is abelian, periodic, and $R/J(R)$ is isomorphic to a field $F$, ${\Bbb Z}_{3}\oplus {\Bbb Z}_{3}$, ${\Bbb Z}_{3}\oplus B$ where $B$ is Boolean, or a Boolean ring. As a specif…

Ring theoretical properties of affine cellular algebras

As a generalisation of Graham and Lehrer's cellular algebras, affine cellular algebras have been introduced in [12] in order to treat affine versions of diagram algebras like affine Hecke algebras of type A and affine Temperley-Lieb algebras in a unifying fashion. Affine cellular algebras include Kleshchev's graded quasihereditary algebras, KLR algebras and various other classes of algebras. In this paper we will study ring theoretical properties of affine cellular algebras. We show that any affine cellular algebra $A$ satisfies a polynomial identity. Furthermore, we show that $A$ can be embed…

图函数的零因子代数：量子捕获、纠缠路由和随机首次通过排除

Zero-divisor algebras of graph functions: quantum caging, entangling routing and stochastic first-passage exclusion

研究利用含零因子交换代数系数在图顶点函数上构造李双代数微分结构，推导雅可比准则并分类其解，证明相关刚性定理，展示该结构在量子捕获、纠缠路由等方面的作用及相关模型的特性。

抽象分解定理及其在幂等分解中的应用

Abstract Factorization Theorems with Applications to Idempotent Factorizations

本文在预序幺半群上建立抽象分解定理，证明阿尔丁性保证通过不可约元分解、强阿尔丁性保证通过夸克分解，并给出最短分解长度上界；特化到环的零因子子幺半群后，恢复并改进了幂等分解的经典定理。