Classification of rings with toroidal Jacobson graph

Czechoslovak Mathematical Journal

Krishnan Selvakumar¹ 

[1] Manoharan Subajini, Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli, Tamil Nadu, 627012, India

关键词: planar graph;    genus of a graph;    local ring;    nilpotent element;    Jacobson graph;   

DOI  :  

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 摘 要 】

Let $R$ be a commutative ring with nonzero identity and $J(R)$ the Jacobson radical of $R$. The Jacobson graph of $R$, denoted by $\mathfrak J_R$, is defined as the graph with vertex set $R\setminus J(R)$ such that two distinct vertices $x$ and $y$ are adjacent if and only if $1-xy$ is not a unit of $R$. The genus of a simple graph $G$ is the smallest nonnegative integer $n$ such that $G$ can be embedded into an orientable surface $S_n$. In this paper, we investigate the genus number of the compact Riemann surface in which $\mathfrak J_R$ can be embedded and explicitly determine all finite commutative rings $R$ (up to isomorphism) such that $\mathfrak J_R$ is toroidal.

【 授权许可】

Unknown