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

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 $$z\in Z(R)\setminus \text{Nil}(R)$$.

### MSC:

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

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