Divisor Graphs of a Commutative Ring

Abstract

If x is an element of a commutative ring R then define the x-divisor graph $\varGamma _x(R)$ to be the graph, whose vertices are the elements of $d(x)=\{r\in R$ | $rs=x$ for some $s\in R\}$ such that two distinct vertices r and s are adjacent if and only if $rs=x$. In this chapter, the components of $\varGamma _x(R)$ are completely characterized when R is a von Neumann regular ring. Various other types of “divisor graphs” are considered as well. For example, if x is a nonzero element of an integral domain R with group of units U(R) then the compressed divisor graph $(\varGamma _E)_x^{d^\times }(R)$ associated with x is defined to be the graph, whose vertices are the associate-equivalence classes $\overline{r}=rU(R)$ of elements $r\in d(x)^\times =d(x)\setminus (xU(R)\cup U(R))$ such that two distinct vertices $\overline{r}$ and $\overline{s}$ are adjacent if and only if $rs\in d(x)$. Alternatively, by letting M be the positive cone of the group of divisibility of R, every $(\varGamma _E)_x^{d^\times }(R)$ is a member of the class of graphs $\varGamma _{\le x}(M)$ defined by picking an element x of a partially ordered commutative monoid M with least element equal to its identity 1, and letting the vertices of $\varGamma _{\le x}(M)$ be the elements of $\{m\in M$ | $1<m<x\}$ such that two distinct vertices m and n are adjacent if and only if $mn\le x$. Other aspects of the chapter include the exploration of graph-theoretic criteria that reveal when two elements of an integral domain are associates, and it is proved that R is a unique factorization domain if and only if $(\varGamma _E)_x^{d^\times }(R)$ is either null or finite with a dominant clique for every $x\in R\setminus \{0\}$. Throughout, emphasis is placed on similarities with zero-divisor graphs. For example, it is proved that if R is von Neumann regular and G is a component of $\varGamma _x(R)$ that contains a square root of x then $G\cong \varGamma _0(\text {ann}_R(x))$ (in particular, if $x=0$ then we have the tautology $G\cong \varGamma _0(R)$), and if x is a square-free element of a unique factorization domain then $(\varGamma _E)_x^{d^\times }(R)$ is isomorphic to a zero-divisor graph of a finite Boolean ring.

This work is dedicated to David Fenimore Anderson, whose guidance and leadership has fostered multitudes of mathematical pursuits, and whose friendship has inspirited a lighthearted approach to all of them.