On the zero-divisor graph of a commutative ring

By Saieed Akbari, Ali Mohammadian. 2004

108 citations recorded by the source corpus.

Abstract Let R be a commutative ring and Γ ( R ) be its zero-divisor graph. In this paper it is shown that for any finite commutative ring R , the edge chromatic number of Γ ( R ) is equal to the maximum degree of Γ ( R ), unless Γ ( R ) is a complete graph of odd order. In [D.F. Anderson, A. Frazier, A. Lauve, P.S. Livingston, in: Lecture Notes in Pure and Appl. Math., Vol. 220, Marcel Dekker, New York, 2001, pp. 61–72] it has been proved that if R and S are finite reduced rings which are not fields, then Γ ( R )≃ Γ ( S ) if and only if R ≃ S . Here we generalize this result and prove that if R is a finite reduced ring which is not isomorphic to Z 2 × Z 2 or to Z 6 and S is a ring such that Γ ( R )≃ Γ ( S ), then R ≃ S .

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b7e2441d50a2e88de93e69c3466d56e9fde965ce

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In Algebra and Number Theory

1. Ali Mohammadian
2. Saieed Akbari

In Geometry and Topology

1. Saieed Akbari
2. Ali Mohammadian

Citations in the corpus

• Zero-divisor graphs of non-commutative rings (2006, background): “r, In [2] it has been proved that for any finite commutative ring R with identity, with finitely many exceptions, ifΓ (R) is isomorphic to the zero-divisor graph of a reduc ring S, thenR S.”