Central sets and radii of the zero-divisor graphs of commutative rings. (English) Zbl 1105.13007

Redmond, Shane P.

Commun. Algebra 34, No. 7, 2389-2401 (2006).

Given a commutative ring \(R\) with identity, the zero-divisor graph of \(R\) is the graph \(\Gamma(R)\) where the vertices are the nonzero zero-divisors, and where there is an undirected edge between two vertices \(x\) and \(y\) if \(xy=0\). This graph and its properties have been examined by a number of different authors.
The diameter of a connected graph \(G\) is the maximum distance between any two distinct vertices. The radius of \(G\) is the minimum of \(\{e(x): x\) a vertex in \(G \}\), where \(e(x)\) denotes the maximum distance from \(x\) to any other vertex of \(G\). The center of \(G\) is the set of vertices \(x\), such that \(e(x)\) equals the radius. It is well known that for any graph \(G\) the diameter is bounded by twice the radius.
It has been shown that for any commutative ring \(R\), if \(\Gamma(R)\) is connected, then the graph has diameter at most 3 and all numbers between 0 and 3 are possible.
The current article gives further descriptions of \(\Gamma(R)\) in terms of the above parameters under certain assumption on \(R\). If \(R\) is noetherian and not an integral domain, then the radius of \(\Gamma(R)\) is at most 2. As a corollary, if \(R\) is noetherian and not an integral domain, then there exist a nonzero \(x\in R\) such that either \(xy=0\) or ann\((x)\cap\) ann\((y)\neq\{0\}\) for all zero-divisors \(y\in R\).
The center of a ring \(R\) can be described (up to several options) in certain cases, such as when \(R\) is an Artinian ring. Furthermore, when \(R\) is Artinian the radius can be determined from the diameter. A dominating set of \(\Gamma(R)\) is constructed using elements of the center when \(R\) is Artinian.

Reviewer: J. Shapiro (Fairfax)

MSC:

13A99	General commutative ring theory
05C99	Graph theory
13M99	Finite commutative rings

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