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Classification of Rings Associated with the Genus of Clean graphs

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ALGEBRA COLLOQUIUM 2024年第3期31卷 451-466页

作者： Ramanathan, V. Selvaraj, C. Altaf, Aaqib Pirzada, S. Periyar Univ Dept Math Salem 636011 Tamil Nadu India Univ Kashmir Dept Math Srinagar Kashmir India

Let R be a ring (not necessarily commutative) with identity 1 and let CL ( R ) be its clean graph. In this paper, we investigate the genus number of the compact Riemann surface in which CL ( R ) can be embedded and explicitly determine all commutative rings R (up to isomorphism) such that CL ( R ) has genus at most two. It is shown that for any Artinian ring R, CL ( R ) is a projective graph if and only if R is isomorphic to F-4 . Furthermore, we determine all isomorphism classes of commutative rings whose clean graphs have crosscap two.

关键词： clean graph commutative ring planar graph genus crosscap zero-divisor graph PLANAR

On Nilpotent Finite Alternative Rings with Planar zero-divisor graphs

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Algebra Colloquium 2016年第4期23卷 657-661页

作者： A.S. Kuzmina Institute of the Physical and Mathematical Education Altai State Pedagogical University

In this paper we prove that any finite nilpotent alternative ring with planar zero-divisor graph is associative.

关键词： zero-divisor graph alternative ring finite ring nilpotent ring planar graph

zero-divisor graphs for Direct Products of Rings

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Chinese Quarterly Journal of Mathematics 2011年第4期26卷 621-627页

作者：李云慧唐高华 Yuan'an No.1 Senior High School Department of Mathematics Guangxi Teacher's College

In [1], Joe Warfel investigated the diameter of a zero-divisor graph for a direct product R 1 × R 2 with respect to the diameter of the zero-divisor graph of R 1 and R 2 . But the author only considered those graphs whose diameters ≥ 1 and discussed six cases. This paper further discusses the other nine cases and also gives a complete characterization for the possible diameters for left Artin rings.

关键词： zero-divisor graph diameter Artin ring local ring

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Undirected zero-divisor graphs and Unique Product Monoid Rings

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Algebra Colloquium 2019年第4期26卷 665-676页

作者： Ebrahim Hashemi Abdollah Alhevaz LABokut Faculty of Mathematical Sciences Shahrood University of Technology P.O.Box:316-***ShahroodIran 不详

Let R be an associative ring with identity and Z^*(K)be its set of non-zero *** undirected zero-divisor graph of R、denoted byΓ(R),is the graph whose vert ices are the non-zero zero-divisors of R、and where two distinct verticesγand s are adjacent if and only ifγs=0 or sγ=*** dist ance bet ween vertices a and b is the length of the shortest path connecting them,and the diameter of the graph,diam(Γ(R)),is the superimum of these *** this paper,first we prove some results aboutΓ(R)of a semi-commutative ring ***,for a reversible ring R and a unique product monoid M、we prove 0≦diam(Γ(R))

关键词： zero-divisor graph diameter semi-commutative ring unique product monoid monoid ring

The zero-divisor graphs of Abelian Regular Rings

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Northeastern Mathematical Journal 2004年第3期20卷 339-348页

作者：卢丹诚佟文廷 Department of Mathematics Suzhou University Suzhou 215006 ) if and only if they have isomorphic reduced zero-divisor graphs. It is shown that the maximal right quotient ring of a potent semiprimitive normal ring is abelian regular moreover the zero-divisor graph of such a ring is studied.

We introduce the zero-divisor graph for an abelian regular ring and show that if R,S are abelian regular, then (K0(R),[R])≌(K0(S),[S]) if and only if they have isomorphic reduced zero-divisor graphs. It is shown that the maximal right quotient ring of a potent semiprimitive normal ring is abelian regular, moreover, the zero-divisor graph of such a ring is studied.

关键词： zero-divisor graph abelian regular ring Grothendieck group

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Quasi-zero-divisor graphs of Non-Commutative Rings

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Journal of Mathematical Research with Applications 2017年第2期37卷 137-147页

作者： Shouxiang ZHAO Jizhu NAN Gaohua TANG School of Mathematical Sciences Dalian University of Technology Department of Mathematics and Computer Science Guilin Normal College School of Mathematical and Statistics Sciences Guangxi Teachers Education University

In this paper, a new class of rings, called FIC rings, is introduced for studying quasi-zero-divisor graphs of rings. Let R be a ring. The quasi-zero-divisor graph of R, denoted by Γ;（R）, is a directed graph defined on its nonzero quasi-zero-divisors, where there is an arc from a vertex x to another vertex y if and only if x Ry = 0. We show that the following three conditions on an FIC ring R are equivalent:（1） χ（R） is finite;（2） ω（R） is finite;（3）Nil;R is finite where Nil;R equals the finite intersection of prime ideals. Furthermore, we also completely determine the connectedness, the diameter and the girth of Γ;（R）.

关键词： quasi-zero-divisor zero-divisor graph chromatic number clique number FIC ring

Ideal-based zero-divisor graphs of Non-commutative Rings

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Chinese Quarterly Journal of Mathematics 2011年第1期26卷 125-130页

作者： LI Yun-hui TANG Gao-hua Yuan＇an No.1 Senior High School Yichang 443300China Department of Mathematics Guangxi Teacher＇s CollegeNanning 530001China

This paper introduces an ideal-boyed zero-divisor graph of non-commutative rings,denoted ΓI（R）.ΓI（R） is a directed *** properties and possible structures of the graph is studied.

关键词： non-commutative ring ideal-based zero-divisor graph diameter

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Ideal-Based k-zero-divisor Hypergraph of Commutative Rings

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Algebra Colloquium 2021年第4期28卷 655-672页

作者： K.Selvakumar M.Subajini Department of Mathematics Manonmaniam Sundaranar University Tirunelveli 627012Tamil NaduIndia

Let R be a commutative ring,I an ideal of R and k≥2 a fixed *** ideal-based k-zero-divisor hypergraph HkI(R)of R has vertex set ZI(R,k),the set of all ideal-based k-zero-divisors of R,and for distinct elements x1,x2,…,xk in ZI(R,k),the set{x1,x2,…,xk}is an edge in HkI(R)if and only if x1x2…xk∈I and the product of the elements of any(k-1)-subset of{x1,x2,…,xk}is not in *** this paper,we show that H3I(R)is connected with diameter at most 4 provided that x^(2)(∈)I for all ideal-based 3-zero-divisor ***,we find the chromatic number of H3(R)when R is a product of finite ***,we find some necessary conditions for a finite ring R and a nonzero ideal I of R to have H3I(R)planar.

关键词： hypergraph zero-divisor graph linear local ring k-prime ideal

On a Class of Semigroup graphs

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Advances in Pure Mathematics 2023年第6期13卷 303-315页

作者： Li Chen Tongsuo Wu School of Mathematics and Statistics Yancheng Teachers University Yancheng China Department of Mathematics Shanghai Jiaotong University Shanghai China

Let G = Γ(S) be a semigroup graph, i.e., a zero-divisor graph of a semigroup S with zero element 0. For any adjacent vertices x, y in G, denote C(x,y) = {z∈V(G) | N (z) = {x,y}}. Assume that in G there exist two adjacent vertices x, y, a vertex s∈C(x,y) and a vertex z such that d (s,z) = 3. This paper studies algebraic properties of S with such graphs G = Γ(S), giving some sub-semigroups and ideals of S. It constructs some classes of such semigroup graphs and classifies all semigroup graphs with the property in two cases.

关键词： zero-divisor Semigroup Sub-Semigroup zero-divisor graph