关于quandle与quandle环的进一步研究
A further study of quandles and quandle rings
浏览论文内容
中文总结 AI 辅助
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。
中文摘要 AI 辅助
我们研究核心quandle以及核心quandle的quandle环中的幂等元，解答了核心quandle的秩、quandle环中非平凡幂等元的若干问题，还解决了近期一篇论文中提出的关于quandle环$\boldsymbol{\boldsymbol{Z}[R_5]}$与$\boldsymbol{\boldsymbol{Z}[C_5]}$中非平凡幂等元的两个问题，其中$\boldsymbol{\boldsymbol{Z}}$是整数环，$\boldsymbol{\boldsymbol{R_5}}$是5阶二面体quandle，$\boldsymbol{\boldsymbol{C_5}}$是5阶交换quandle。随后，我们在基环为整环的情况下，研究平凡quandle与Joyce quandle的扩展quandle环中的单位元，以及平凡quandle的扩展quandle环中的幂零元，由此证明平凡quandle的quandle环与扩展quandle环均不是幂零clean环。我们还探究了quandle环中的素环与半素环，引入quandle环的零因子图，发现顶点入度与出度间存在有趣的镜像对称性，此外，在环$\boldsymbol{\boldsymbol{(Z_{2n+1}[Q])[X,Y]}}$中找到一个二元多项式，可确定$\boldsymbol{\boldsymbol{2n+1}}$阶交换quandle，其中$\boldsymbol{\boldsymbol{2n+1}}$为素数。
英文摘要
We investigate core quandles and idempotents in quandle rings of core quandles. We answer several questions on the rank of core quandles and nontrivial idempotents in quandle rings. We also present solutions to two questions raised in a recent paper about non-trivial idempotents in quandle rings $\mathbb{Z}[R_5]$ and $\mathbb{Z}[C_5]$, where $\mathbb{Z}$, $R_5$ and $C_5$ are the ring of integers, the dihedral quandle of order 5, and the commutative quandle of order 5, respectively. We then study units in extended quandle rings of a trivial quandle and the Joyce quandle, and nilpotent elements in extended quandles rings of a trivial quandle, where the ground ring is an integral domain. As a consequence, we show that the quandle ring and the extended quandle ring of a trivial quandle are not nil clean rings. We also explore prime rings and semi-prime rings among quandle rings. We introduce zero-divisor graphs of quandle rings and find an intriguing mirror symmetry among the in-degree and out-degree of the vertices. Moreover, we find a bivariate polynomial in the ring $(\mathbb{Z}_{2n+1}[Q])[X,Y]$ that determines the commutative quandle of order $2n+1$, where $2n+1$ is prime.
