On the genus of reduced cozero-divisor graph of commutative rings

被引：0
作者：
E. Jesili
K. Selvakumar
T. Tamizh Chelvam
机构：
[1] Manonmaniam Sundaranar University,Department of Mathematics
来源：
Soft Computing | 2023年 / 27卷
关键词：
Artinian ring; Reduced co-zero divisor graph; Genus of a graph; Planar graph;
D O I：
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摘要：
Let R be a commutative ring with identity and let (x) be the principal ideal generated by x∈R.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in R.$$\end{document} Let Ω(R)∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega (R)^*$$\end{document} be the set of all nontrivial principal ideals of R. The reduced cozero-divisor graph Γr(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varGamma _r(R)$$\end{document} of R is an undirected simple graph with Ω(R)∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega (R)^*$$\end{document} as the vertex set and two distinct vertices (x) and (y) in Ω(R)∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega (R)^*$$\end{document} are adjacent if and only if (x)⊈(y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x)\nsubseteq (y)$$\end{document} and (y)⊈(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(y)\nsubseteq (x)$$\end{document}. In this paper, we characterize all classes of commutative Artinian non-local rings for which the reduced cozero-divisor graph has genus at most one.
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页码：657 / 666
页数：9
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