We introduce the concepts of left （right） zero-divisor rings, a class of rings without identity. We call a ring R left （right） zero-divisor if rR（a） ≠ 0（lR（a） ≠ 0） for every a∈ R, and call R strong left ...We introduce the concepts of left （right） zero-divisor rings, a class of rings without identity. We call a ring R left （right） zero-divisor if rR（a） ≠ 0（lR（a） ≠ 0） for every a∈ R, and call R strong left （right） zero-divisor if r R （R）≠0（lR（R）≠ 0）. Camillo and Nielson called a ring right finite annihilated （RFA） if every finite subset has non-zero right annihilator. We present in this paper some basic examples of left zero-divisor rings, and investigate the extensions of strong left zero-divisor rings and RFA rings, giving their equivalent characterizations.展开更多
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 Г＊（R）, is a directed graph defin...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 Г＊（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 xRy = 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 Г＊ （R）.展开更多
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 gra...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.展开更多
This paper introduces an ideal-boyed zero-divisor graph of non-commutative rings,denoted ΓI（R）.ΓI（R） is a directed graph.The properties and possible structures of the graph 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...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.展开更多
A graph is called a proper refinement of a star graph if it is a refinement of a star graph, but it is neither a star graph nor a complete graph. For a refinement of a star graph G with center c, let G＊ be the subgra...A graph is called a proper refinement of a star graph if it is a refinement of a star graph, but it is neither a star graph nor a complete graph. For a refinement of a star graph G with center c, let G＊ be the subgraph of G induced on the vertex set V（G） ／ {c or end vertices adjacent to c}. In this paper, we study the isomorphic classification of some finite commutative local rings R by investigating their zero-divisor graphs G=Г（R）, which is a proper refinement of a star graph with exactly one center c. We determine all finite commutative local rings R such that G＊ has at least two connected components. We prove that the diameter of the induced graph G＊ is two if Z（R）2 ≠{0}, Z（R）3 = {0} and Gc is connected. We determine the structure of R which has two distinct nonadjacent vertices a, fl C Z（R）＊／{c} such that the ideal [N（a）N（β）]{0} is generated by only one element of Z（R）＊／{c}. We also completely determine the correspondence between commutative rings and finite complete graphs Kn with some end vertices adjacent to a single vertex of Kn.展开更多
Let R be a commutative ring,I an ideal of R and k≥2 a fixed integer.The 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 element...Let R be a commutative ring,I an ideal of R and k≥2 a fixed integer.The 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 I.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 hypergraphs.Moreover,we find the chromatic number of H3(R)when R is a product of finite fields.Finally,we find some necessary conditions for a finite ring R and a nonzero ideal I of R to have H3I(R)planar.展开更多
Let R be an associative ring with identity and Z^*(K)be its set of non-zero zero-divisors.The undirected zero-divisor graph of R、denoted byΓ(R),is the graph whose vert ices are the non-zero zero-divisors of R、and w...Let R be an associative ring with identity and Z^*(K)be its set of non-zero zero-divisors.The 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=0.The 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 distances.In this paper,first we prove some results aboutΓ(R)of a semi-commutative ring R.Then,for a reversible ring R and a unique product monoid M、we prove 0≦diam(Γ(R))<diam(Γ(R[M]))≦3.We describe all the possibilities for the pair diam(Γ(R))and diam(Γ(R[M])),strictly in terms of the properties of a ring R,where K is a reversible ring and M is a unique product monoid.Moreover,an example showing the necessity of our assumptions is provided.展开更多
Let Pn be a path graph with n vertices, and let Fn = Pn ∪ {c}, where c is adjacent to all vertices of Pn. The resulting graph is called a fan-shaped graph. The corresponding zero-divisor semigroups have been complete...Let Pn be a path graph with n vertices, and let Fn = Pn ∪ {c}, where c is adjacent to all vertices of Pn. The resulting graph is called a fan-shaped graph. The corresponding zero-divisor semigroups have been completely determined by Tang et al. for n = 2, 3, 4 and by Wu et al. for n ≥ 6, respectively. In this paper, we study the case for n = 5, and give all the corresponding zero-divisor semigroups of Fn.展开更多
This research investigates the comparative efficacy of generating zero divisor graphs (ZDGs) of the ring of integers ℤ<sub>n</sub> modulo n using MAPLE algorithm. Zero divisor graphs, pivotal in the study ...This research investigates the comparative efficacy of generating zero divisor graphs (ZDGs) of the ring of integers ℤ<sub>n</sub> modulo n using MAPLE algorithm. Zero divisor graphs, pivotal in the study of ring theory, depict relationships between elements of a ring that multiply to zero. The paper explores the development and implementation of algorithms in MAPLE for constructing these ZDGs. The comparative study aims to discern the strengths, limitations, and computational efficiency of different MAPLE algorithms for creating zero divisor graphs offering insights for mathematicians, researchers, and computational enthusiasts involved in ring theory and mathematical computations.展开更多
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 exi...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.展开更多
基金Supported by the National Natural Science Foundation of China(Grant Nos.1107109711101217)
文摘We introduce the concepts of left （right） zero-divisor rings, a class of rings without identity. We call a ring R left （right） zero-divisor if rR（a） ≠ 0（lR（a） ≠ 0） for every a∈ R, and call R strong left （right） zero-divisor if r R （R）≠0（lR（R）≠ 0）. Camillo and Nielson called a ring right finite annihilated （RFA） if every finite subset has non-zero right annihilator. We present in this paper some basic examples of left zero-divisor rings, and investigate the extensions of strong left zero-divisor rings and RFA rings, giving their equivalent characterizations.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1137134311161006+4 种基金1166101411171142)the Guangxi Science Research and Technology Development Project(Grant No.1599005-2-13)the Scientic Research Fund of Guangxi Education Department(Grant No.KY2015ZD075)the Natural Science Foundation of Guangxi(Grant No.2016GXSFDA380017)
文摘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 Г＊（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 xRy = 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 Г＊ （R）.
基金Supported by the Natural Sciences Foundation of Guangxi Province(0575052, 0640070)Supported by the Innovation Project of Guangxi Graduate Education(2006106030701M05)Supported by the Scientific Research Foundation of Guangxi Educational Committee(200707LX233
文摘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.
基金Supported by Guangxi Natural Sciences Foundation(0575052,0640070)Supported byInnovation Project of Guangxi Graduate Education(2006106030701M05)Supported Scientific Research Foun-dation of Guangxi Educational Committee
文摘This paper introduces an ideal-boyed zero-divisor graph of non-commutative rings,denoted ΓI（R）.ΓI（R） is a directed graph.The properties and possible structures of the graph is studied.
基金Partially supported by the NSF (10071035) of China.
文摘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.
基金Supported by National Natural Science Foundation of China （Grant No. 10671122） the first author is supported by Youth Foundation of Shanghai （Grant No. sdl10017） and also partly supported by Natural Science Foundation of Shanghai （Grant No. 10ZR1412500） the second author is partly supported by STCSM （Grant No. 09XD1402500）
文摘A graph is called a proper refinement of a star graph if it is a refinement of a star graph, but it is neither a star graph nor a complete graph. For a refinement of a star graph G with center c, let G＊ be the subgraph of G induced on the vertex set V（G） ／ {c or end vertices adjacent to c}. In this paper, we study the isomorphic classification of some finite commutative local rings R by investigating their zero-divisor graphs G=Г（R）, which is a proper refinement of a star graph with exactly one center c. We determine all finite commutative local rings R such that G＊ has at least two connected components. We prove that the diameter of the induced graph G＊ is two if Z（R）2 ≠{0}, Z（R）3 = {0} and Gc is connected. We determine the structure of R which has two distinct nonadjacent vertices a, fl C Z（R）＊／{c} such that the ideal [N（a）N（β）]{0} is generated by only one element of Z（R）＊／{c}. We also completely determine the correspondence between commutative rings and finite complete graphs Kn with some end vertices adjacent to a single vertex of Kn.
基金supported by the SERB-EEQ project(EEQ/2016/000367)of Department of Science and Technology,Indiasupported by the INSPIRE programme(IF 140700)of Department of Science and Technology,India.
文摘Let R be a commutative ring,I an ideal of R and k≥2 a fixed integer.The 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 I.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 hypergraphs.Moreover,we find the chromatic number of H3(R)when R is a product of finite fields.Finally,we find some necessary conditions for a finite ring R and a nonzero ideal I of R to have H3I(R)planar.
文摘Let R be an associative ring with identity and Z^*(K)be its set of non-zero zero-divisors.The 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=0.The 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 distances.In this paper,first we prove some results aboutΓ(R)of a semi-commutative ring R.Then,for a reversible ring R and a unique product monoid M、we prove 0≦diam(Γ(R))<diam(Γ(R[M]))≦3.We describe all the possibilities for the pair diam(Γ(R))and diam(Γ(R[M])),strictly in terms of the properties of a ring R,where K is a reversible ring and M is a unique product monoid.Moreover,an example showing the necessity of our assumptions is provided.
基金Supported by the Guangxi Natural Science Foundation (Grant Nos.2010GXNSFB0130480991102+1 种基金2011GXNSFA018139)the Scientific Research Foundation of Guangxi Educational Committee (Grant No. 200911LX275)
文摘Let Pn be a path graph with n vertices, and let Fn = Pn ∪ {c}, where c is adjacent to all vertices of Pn. The resulting graph is called a fan-shaped graph. The corresponding zero-divisor semigroups have been completely determined by Tang et al. for n = 2, 3, 4 and by Wu et al. for n ≥ 6, respectively. In this paper, we study the case for n = 5, and give all the corresponding zero-divisor semigroups of Fn.
文摘This research investigates the comparative efficacy of generating zero divisor graphs (ZDGs) of the ring of integers ℤ<sub>n</sub> modulo n using MAPLE algorithm. Zero divisor graphs, pivotal in the study of ring theory, depict relationships between elements of a ring that multiply to zero. The paper explores the development and implementation of algorithms in MAPLE for constructing these ZDGs. The comparative study aims to discern the strengths, limitations, and computational efficiency of different MAPLE algorithms for creating zero divisor graphs offering insights for mathematicians, researchers, and computational enthusiasts involved in ring theory and mathematical computations.
文摘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.
