The idempotent-divisor graphs of a commutative ring

ABSTRACT

Given an idempotent element e of a commutative ring R, define the idempotent-divisor graph Γ_e(R) of R associated with e to be the (undirected) graph with vertex-set V(Γ_e(R)) = {a∈R | there exists b∈R with ab = e} such that distinct a,b∈V(Γ_e(R)) are adjacent if and only if ab = e. In this paper, the interplay between the algebraic properties of R and the graph-theoretic structure of Γ_e(R) is investigated. For example, it is shown that if Re is a Boolean ring then Γ_e(R) is connected, and if R has identity then Γ_e(R) is finite if and only if R(1−e) and U(R) are finite, where U(R) is the group of units of R. Furthermore, if e∈R and Γ_e(T(R)) is connected then ${\Gamma }_e\left(R\right)\cong {\Gamma }_e\left(T\right(R\left)\right)$, where T(R) is the total quotient ring of R. Emphasis is given to linking the structure of Γ_e(R) with that of zero-divisor graphs (i.e., idempotent-divisor graphs associated with 0).

KEYWORDS:

• Boolean ring
• commutative ring
• idempotent-divisor graph
• zero-divisor graph

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 05C25, 13A99, 06E99

Acknowledgments

Part of the research presented in this article is the result of participation by the first author (C.F. Kimball) in a special topics course on commutative rings during Spring, 2016, and an independent study during the 2016–2017 academic year given at Lindsey Wilson College by the second author (J.D. LaGrange). The authors wish to express their gratitude to Amanda K. Gerald for her contributions to several fruitful discussions which helped initiate the some of the investigations in this article.