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**On the zero-divisor graph of a commutative ring.**
*(English)*
Zbl 1085.13011

From the paper: Throughout the paper, all rings are assumed to be commutative with unity \(1\neq 0\). If \(R\) is a ring, \(Z(R)\) denotes its set of zero-divisors. A ring \(R\) is said to be reduced if \(R\) has no non-zero nilpotent element. The zero-divisor graph of \(R\), denoted by \(\Gamma (R)\), is a graph with vertex set \(Z(R)\setminus\{0\}\) in which two vertices \(x\) and \(y\) are adjacent if and only if \(x\neq y\) and \(xy=0\). A \(k\)-edge coloring of a graph \(G\) is an assignment of colors \(\{1,\dots, k\}\) to the edges of \(G\) such that no two adjacent edges have the same color. The edge chromatic number \(\chi'(G)\) of a graph \(G\) is the minimum \(k\) for which \(G\) has a \(k\)-edge coloring. In the paper under review, it is shown that for any finite commutative ring \(R\), the edge chromatic number of \(\Gamma(R)\) is equal to the maximum number of edges of \(\Gamma(R)\), unless \(\Gamma(R)\) is a complete graph of odd order.

D. F. Anderson, A. Frazier, A. Lauve and P. S. Livingston [in: Ideal theoretic methods in commutative algebra, Lect. Notes Pure Appl. Math. 220, 61–72 (2001; Zbl 1035.13004)] proved that if \(R\) and \(S\) are finite reduced rings which are not fields, then \(\Gamma(R)\simeq\Gamma(S)\) if and only if \(R\simeq S\). Here we generalize this result and prove that if \(R\) is a finite reduced ring which is not isomorphic to \(\mathbb{Z}_2\times\mathbb{Z}_2\) or to \(\mathbb{Z}_6\) and \(S\) is a ring such that \(\Gamma(R)\simeq\Gamma(S)\), then \(R\simeq S\).

D. F. Anderson, A. Frazier, A. Lauve and P. S. Livingston [in: Ideal theoretic methods in commutative algebra, Lect. Notes Pure Appl. Math. 220, 61–72 (2001; Zbl 1035.13004)] proved that if \(R\) and \(S\) are finite reduced rings which are not fields, then \(\Gamma(R)\simeq\Gamma(S)\) if and only if \(R\simeq S\). Here we generalize this result and prove that if \(R\) is a finite reduced ring which is not isomorphic to \(\mathbb{Z}_2\times\mathbb{Z}_2\) or to \(\mathbb{Z}_6\) and \(S\) is a ring such that \(\Gamma(R)\simeq\Gamma(S)\), then \(R\simeq S\).

### MSC:

13M05 | Structure of finite commutative rings |

13A05 | Divisibility and factorizations in commutative rings |

05C15 | Coloring of graphs and hypergraphs |

05C90 | Applications of graph theory |

13F55 | Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes |

### Keywords:

zero-divisor graph; edge coloring; Hamiltonian graph; edge chromatic number; finite commutative ring### Citations:

Zbl 1035.13004
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\textit{S. Akbari} and \textit{A. Mohammadian}, J. Algebra 274, No. 2, 847--855 (2004; Zbl 1085.13011)

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### References:

[1] | S. Akbari, H.R. Maimani, S. Yassemi, When a zero-divisor graph is planar or complete r-partite graph, J. Algebra, submitted for publication · Zbl 1032.13014 |

[2] | Anderson, D.F.; Frazier, A.; Lauve, A.; Livingston, P.S., The zero-divisor graph of a commutative ring, II, (), 61-72 · Zbl 1035.13004 |

[3] | D.F. Anderson, R. Levy, J. Shapiro, Zero-divisor graphs, von Neumann regular rings, and Boolean algebras, J. Pure Appl. Algebra, submitted for publication · Zbl 1076.13001 |

[4] | Anderson, D.F.; Livingston, P.S., The zero-divisor graph of a commutative ring, J. algebra, 217, 434-447, (1999) · Zbl 0941.05062 |

[5] | Atiyah, M.F.; Macdonald, I.G., Introduction to commutative algebra, (1969), Addison-Wesley Reading, MA · Zbl 0175.03601 |

[6] | Beck, I., Coloring of commutative rings, J. algebra, 116, 208-226, (1988) · Zbl 0654.13001 |

[7] | Redmond, S.P., The zero-divisor graph of a non-commutative ring, Internat. J. commutative rings, 1, 4, 203-211, (2002) · Zbl 1195.16038 |

[8] | Yap, H.P., Some topics in graph theory, London math. soc. lecture note ser., vol. 108, (1986) · Zbl 0588.05002 |

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