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The Comaximal Graphs of Noncommutative Rings

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For a ring R(not necessarily commutative)with identity,the comaximal graph of R,denoted by Ω(R),is a graph whose vertices are all the nonunit elements of R,and two distinct vertices a and b are adjacent if and only if Ra+Rb=R.In this paper we consider a subgraph Ω1(R)of S(R)induced by R\Uℓ(R),where Uℓ(R)is the set of all left-invertible elements of R.We characterize those rings R for which Ω1(R)\J(R)is a complete graph or a star graph,where J(R)is the Jacobson radical of R.We investigate the clique number and the chromatic number of the graph Q1(R)\ J(R),and we prove that if every left ideal of R is symmetric,then this graph is connected and its diameter is at most 3.Moreover,we completely characterize the diameter of Ω1(R)\J(R).We also investigate the properties of R when Ω1(R)is a split graph.

comaximal graphnoncommutative ringleft invertible elementsplit graph

Shouqiang Shen、Weijun Liu、Lihua Feng

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School of Applied Science,Beijing Information Science and Technology University Beijing 100192,China

School of Mathematics and Statistics, HNP-LAMA Central South University, Changsha 410083, China

NSFCNSFCHunan Provincial Natural Science FoundationHunan Provincial Natural Science FoundationResearch Fund of Beijing Information Science and Technology University

12071484118714792020JJ46752018JJ24792025030

2023

10.1142/S1005386723000366

代数集刊(英文版)

代数集刊(英文版)

CSCD北大核心
ISSN：1005-3867
年,卷(期)：2023.30(3)
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