Classification of rings with toroidal Jacobson graph

Krishnan Selvakumar; Manoharan Subajini

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 2, page 307-316
  • ISSN: 0011-4642

Abstract

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Let R be a commutative ring with nonzero identity and J ( R ) the Jacobson radical of R . The Jacobson graph of R , denoted by 𝔍 R , is defined as the graph with vertex set R ∖ J ( R ) such that two distinct vertices x and y are adjacent if and only if 1 - x y 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 𝔍 R can be embedded and explicitly determine all finite commutative rings R (up to isomorphism) such that 𝔍 R is toroidal.

How to cite

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Selvakumar, Krishnan, and Subajini, Manoharan. "Classification of rings with toroidal Jacobson graph." Czechoslovak Mathematical Journal 66.2 (2016): 307-316. <http://eudml.org/doc/280102>.

@article{Selvakumar2016,
abstract = {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.},
author = {Selvakumar, Krishnan, Subajini, Manoharan},
journal = {Czechoslovak Mathematical Journal},
keywords = {planar graph; genus of a graph; local ring; nilpotent element; Jacobson graph},
language = {eng},
number = {2},
pages = {307-316},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Classification of rings with toroidal Jacobson graph},
url = {http://eudml.org/doc/280102},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Selvakumar, Krishnan
AU - Subajini, Manoharan
TI - Classification of rings with toroidal Jacobson graph
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 2
SP - 307
EP - 316
AB - 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.
LA - eng
KW - planar graph; genus of a graph; local ring; nilpotent element; Jacobson graph
UR - http://eudml.org/doc/280102
ER -

References

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