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Divisor Graphs of a Commutative Ring

  • John D. LaGrangeEmail author
Chapter
Part of the Trends in Mathematics book series (TM)

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.

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Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Lindsey Wilson CollegeColumbiaUSA

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