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Keywords:
planar graph; genus of a graph; local ring; nilpotent element; Jacobson graph
planar graph; genus of a graph; local ring; nilpotent element; Jacobson graph
Summary:
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.
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