Let R be a commutative ring and Γ(R)be its zero-divisor graph.We completely determine the structure of all finite commutative rings whose zero-divisor graphs have clique number one,two,or three.Furthermore,if R■R1&#...Let R be a commutative ring and Γ(R)be its zero-divisor graph.We completely determine the structure of all finite commutative rings whose zero-divisor graphs have clique number one,two,or three.Furthermore,if R■R1×R2×…×Rn(each Ri is local for i=1,2,3,...,n),we also give algebraic characterizations of the ring R when the clique number of Γ(R)is four.展开更多
We study the algebraic structure of rings R whose zero-divisor graph T(R)has clique number four.Furthermore,we give complete characterizations of all the finite commutative local rings with clique number 4.
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
基金This research was supported by the National Natural Science Foundation of China(No.11801356,No.11401368,No.11971338)by the Natural Science Foundation of Shanghai(No.19ZR1424100).
文摘Let R be a commutative ring and Γ(R)be its zero-divisor graph.We completely determine the structure of all finite commutative rings whose zero-divisor graphs have clique number one,two,or three.Furthermore,if R■R1×R2×…×Rn(each Ri is local for i=1,2,3,...,n),we also give algebraic characterizations of the ring R when the clique number of Γ(R)is four.
基金This research was supported by the National Natural Science Foundation of China(No.11801356,No.11401368,No.11971338)by the Natural Science Foundation of Shanghai(No.19ZR1424100).
文摘We study the algebraic structure of rings R whose zero-divisor graph T(R)has clique number four.Furthermore,we give complete characterizations of all the finite commutative local rings with clique number 4.
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.
