Generalized Irreducible Divisor Graphs

Abstract

In 1988, Beck introduced the notion of a zero-divisor graph of a commutative rings with 1. There have been several generalizations in recent years. In particular, in 2007 Coykendall and Maney developed the irreducible divisor graph. Much work has been done on generalized factorization, especially -factorization. The goal of this paper is to synthesize the notions of -factorization and irreducible divisor graphs in domains. We will define a -irreducible divisor graph for nonzero non unit elements of a domain. We show that, by studying -irreducible divisor graphs, we find equivalent characterizations of several finite -factorization properties.

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

“…We summarize these remarks in the following theorem. Moreover, in an integral domain D with τ a symmetric relation on D # , we get a correspondence between the graphs studied in this paper and those studied in [22].…”

“…In particular, for non-zero, non-units, the set of τ -atoms in [22] is the same as the set of τ -atoms, τ -strong atoms, τ -m-atoms, τ -unrefinable atoms, and τ -very strong atoms as defined in [24] and hence the present article. We also recall that all the associate relations coincide in an integral domain.…”

“…, the τ -irreducible divisor graph of x as in [22]. Furthermore, τ -G β α (x) = G τ (x), the reduced graphs of the preceding statement.…”

“…Recently, the author was able to extend many of these generalized factorization techniques in rings with zero-divisors in [21,23,24,25,27]. In [22], the author extended irreducible divisor graphs by way of τ -factorization in domains.…”

“…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].…”