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 \(R\simeq S\).


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


Zbl 1035.13004
Full Text: DOI


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