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
In this paper we first prove that a near-ring admits a derivation if and only if it is zero-symmetric. Also, we prove some commutativity theorems for a non-necessarily 3-prime near-ring R with a suitably-constrained d...In this paper we first prove that a near-ring admits a derivation if and only if it is zero-symmetric. Also, we prove some commutativity theorems for a non-necessarily 3-prime near-ring R with a suitably-constrained derivation d satisfying the condition that d（a） is not a left zero-divisor in R for some a ∈ R. As consequences, we generalize several commutativity theorems for 3-prime near-rings admitting derivations.展开更多
基金National Natural Science Foundation of China(1107109711101217)Natural Science Foundation of the Education Department of Jiangsu Province(11KJB110007)
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.
文摘In this paper we first prove that a near-ring admits a derivation if and only if it is zero-symmetric. Also, we prove some commutativity theorems for a non-necessarily 3-prime near-ring R with a suitably-constrained derivation d satisfying the condition that d（a） is not a left zero-divisor in R for some a ∈ R. As consequences, we generalize several commutativity theorems for 3-prime near-rings admitting derivations.
