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A zero divisor graph determined by equivalence classes of zero divisors. (English) Zbl 1225.13007
This paper continues with the research in the area of zero divisor graphs, namely the investigation of the interplay between the ring-theoretic properties of a ring \(R\) and the graph theoretic properties of certain graphs obtained from \(R\). The authors focus on \(\Gamma_E(R)\), the zero divisor graph determined by equivalence classes, that is introduced by S. B. Mulay [Commun. Algebra 30, No. 7, 3533–3558 (2002; Zbl 1087.13500)].
The authors consider infinite graphs and star graphs and answer the question of whether or not the Noetherian condition on \(R\) is enough to force \(\Gamma_E(R)\) to be finite. In addition, the relation between the associated primes of \(R\) and the vertices of \(\Gamma_E(R)\) is given. In particular, the authors demonstrate how to identify some elements of the set of associated primes of \(R\).

MSC:
13A15 Ideals and multiplicative ideal theory in commutative rings
13A99 General commutative ring theory
05C12 Distance in graphs
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