This article surveys the recent and active area of zero-divisor graphs of commutative rings. Notable algebraic and graphical results are given, followed by a historical overview and an extensive bibliography.

Zero-divisor graphs in commutative rings

The zero-divisor graph of a commutative ring

This article examines the connections between the factorization properties of a domain, e.g., unique factorization domain (UFD), finite factorization domain (FFD), and the domain's irreducible divisor graphs. In particular, we show that although there are some nice correlations between the properties of the domain D and the set of irreducible divisor graphs {G(x): x ∈ D* \ U(D)} when D is an FFD, it is very unlikely that any information about the domain D can be gleaned from the collection {G(x): x ∈ D* \ U(D)} when D is not an FFD. We also introduce an alternate irreducible divisor graph called the compressed irreducible divisor graph and study some of its properties.

Irreducible Divisor Graphs and Factorization Properties of Domains

In their article “Irreducible Divisor Graphs”, Coykendall and Maney (2007) introduced the idea of irreducible divisor graphs of elements of a domain. We generalize this concept to commutative rings with zero divisors. In particular, the interplay of unique factoring and connected/complete graphs is explored. The diameter and girth of such graphs are also briefly discussed.

Irreducible divisor graphs in commutative rings with zero divisors

Let R be an integral domain, and let x ∈ R be a nonzero nonunit that can be written as a product of irreducibles. Coykendall and Maney (to appear), defined the irreducible divisor graph of x, denoted G(x), as follows. The vertices of G(x) are the nonassociate irreducible divisors of x (each from a pre-chosen coset of the form π U(R) for π ∈ R irreducible). Given distinct vertices y and z, we put an edge between y and z if and only if yz|x. Finally, if y n |x but y n+1 ∤ x, then we put n − 1 loops on the vertex y. In this article, inspired by the approach of the authors from Akhtar and Lee (to appear), we study G(x) using homology. A connection is found between H 1 and the cycle space of G(x). We also characterize UFDs via these homology groups.

Irreducible Divisor Graphs II

In Beck (1988), the author introduces the idea of a zero-divisor graph of a commutative ring. We generalize this idea to study factorization in integral domains and define irreducible divisor graphs. We use these irreducible divisor graphs to characterize certain classes of domains, including UFDs.

Jack Maney

Papers

Irreducible Divisor Graphs

Irreducible Divisor Graphs II