On irreducible divisor graphs in commutative rings with zero-divisors

Abstract

Abstract. In this paper, we continue the program initiated by I. Beck's now classical paper concerning zero-divisor graphs of commutative rings. After the success of much research regarding zero-divisor graphs, many authors have turned their attention to studying divisor graphs of non-zero elements in the ring, the so called irreducible divisor graph. In this paper, we construct several different associated irreducible divisor graphs of a commutative ring with unity using various choices for the definition of irreducible and atomic in the literature. We continue pursuing the program of exploiting the interaction between algebraic structures and associated graphs to further our understanding of both objects. Factorization in rings with zero-divisors is considerably more complicated than integral domains; however, we find that many of the same techniques can be extended to rings with zero-divisors. This allows us to not only find graph theoretic characterizations of many of the finite factorization properties that commutative rings may possess, but also understand graph theoretic properties of graphs associated with certain commutative rings satisfying nice factorization properties. 2010

Τ-Irreducible DIVISOR GRAPHS IN COMMUTATIVE RINGS WITH ZERO-DIVISORS

“…We then summarize many graph theory definitions as well as definitions arising from [10] and [26] in the study of irreducible divisor graphs in rings with zero-divisors.…”

“…In Section 6, we prove several analogous theorems to [26] which illustrate how τ -irreducible divisor graphs give us alternative characterizations of the various τ -finite factorization properties rings may possess as defined in [24].…”

“…Section 2 provides the background information and definitions from the study of irreducible divisor graphs in rings with zero-divisors primarily from [26] as well as τ -factorization in rings with zero-divisors from [24]. In Section 3, we define a variety of τ -irreducible divisor graphs of a commutative ring R given a fixed symmetric and associate preserving relation τ on the non-zero, non-units of R.…”

“…Using irreducible divisor graphs to study finite factorization properties with the usual factorization was carried out in [26] and thus we look to extend this approach to the generalized factorization techniques. We find that many equivalent characterizations of τ -finite factorization properties of commutative rings with zero-divisors given in the aforementioned papers can be provided by studying τ -irreducible divisor graphs.…”

“…In 1988, Beck in [11], introduced the zero-divisor graph, Γ(R) P. Redmond, J. Stickles, A. Lauve, P. S. Livingston, M. Naseer and the author in [3,5,6,7,8,22,24,26,28].…”