Zero divisor graph of a commutative ring

Posted by Jason Polak on 18. January 2018 · Write a comment · Categories: commutative-algebra · Tags: ,

Let $R$ be a commutative ring. The zero divisors of $R$, which we denote $Z(R)$ is the set-theoretic union of prime ideals. This is just because in any commutative ring, the set of subsets of $R$ that can be written as unions of prime ideals is in bijection with the saturated multiplicatively closed sets (the multiplicatively closed sets that contain the divisors of each of their elements).

Istvan Beck in 1986* introduced an undirected graph (in the sense of vertices and edges) associated to the zero divisors in a commutative ring. Recall that an undirected graph is just a set of vertices (points) and edges connecting the point. What is his graph? His idea was to let the vertices correspond to points of $R$, and the edges correspond to the relation than the product of the corresponding elements is zero.

There is a slightly different definition due to Anderson and Livingston, which is the main one used today. Let $Z(R)^*$ denote the nonzero zero divisors. Their graph is $\Gamma(R)$, which is defined as the graph whose vertices are the elements of $Z(R)^*$, and whose edges are defined by connecting two distinct points if and only if their product is zero. Naturally, if $Z(R)^*$ is not empty then the resulting graph $\Gamma(R)$ will have some edges. The actual information contained in $\Gamma(R)$ is pretty much the same as the information contained in Beck’s version and so we’ll just stick with $\Gamma(R)$.

For this idea to be more than just a curiosity, the graph theoretic properties of $\Gamma(R)$ should tell us something about hte ring theoretic properties of $R$. Does it? Anderson and Livingston showed in 1998 that there exists a vertex of $\Gamma(R)$ adjacent to every other vertex if and only if either $R = \Z/2\times A$ where $A$ is an integral domain or $Z(R)$ is an annihilator. They also showed that for $R$ a finite commutative ring, if $\Gamma(R)$ is complete, then $R\cong \Z/2\times \Z/2$ or $R$ is local with characteristic $p$ or $p^2$.
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