On the zero-divisor graph of a ring. (English) Zbl 1152.13001

Anderson, David F.; Badawi, Ayman

Commun. Algebra 36, No. 8, 3073-3092 (2008).

Let \(R\) be a commutative ring with nonzero identity, and let \(Z(R)\) be its set of zero-divisors. The zero-divisor graph, \(\Gamma(R)\), is the graph with vertices the set of nonzero zero-divisors of \(R\), and for distinct \(x,y \in {Z(R)}\), the vertices \(x\) and \(y\) are adjacent if and only if \(xy=0\). The zero-divisor graph of a commutative ring has been studied extensively by several authors.
In this paper the authors study \(\Gamma(R)\) for several classes of rings which generalize valuation domains to the context of rings with zero divisors. These are rings with nonzero zero-divisors that satisfy certain divisibility conditions between elements or comparability conditions between ideals or prime ideals. These rings include chained rings, rings R whose prime ideals contained in \(Z(R)\) are linearly ordered, and rings \(R\) such that \(\{0\}\neq \text{Nil}(R)\subseteq zR\) for all \(z\in Z(R)\setminus \text{Nil}(R)\).

Reviewer: Siamak Yassemi (Tehran)

MSC:

13A15	Ideals and multiplicative ideal theory in commutative rings
13F99	Arithmetic rings and other special commutative rings
05C99	Graph theory

Keywords:

chained rings; linearly ordered prime ideals; zero divisor graph

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