Irreducible Divisor Graphs in Commutative Rings with Zero Divisors

Axtell, Michael; Stickles, Joe

doi:10.1080/00927870801941473

Cited by 12 publications

(24 citation statements)

References 9 publications

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“…This topic has been further studied in [7] and generalized to non-domains in [2]. Given an atomic domain D and a nonzero nonunit x ∈ D, the irreducible divisor graph of x in D, denoted G x has as vertices one representative from each associate class of irreducible divisors of x.…”

Section: Motivation and Terminologymentioning

confidence: 99%

Irreducible Divisor Graphs and Factorization Properties of Domains

Axtell

Baeth

Stickles

2011

Communications in Algebra

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Section: Motivation and Terminologymentioning

confidence: 99%

Irreducible Divisor Graphs and Factorization Properties of Domains

Axtell

Baeth

Stickles

2011

Communications in Algebra

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“…But this again yields two distinct -atomic factorizations of y a 1 , y a 1 = b 1 c 1 · · · c m = a 2 · · · a n again contradicting the minimality of n. Hence, must be empty as desired, completing the proof. (6) For all x ∈ D # , degl a < for all a ∈ V G x ; (7) For all x ∈ D # , deg a < for all a ∈ V G x .…”

Section: The -Irreducible Divisor Graph and -Finite Factorization Promentioning

confidence: 99%

“…Instead of looking exclusively at divisors of zero in a ring, they restrict to a domain D and choose any nonzero, non unit x ∈ D and study the relationships between the irreducible divisors of x. In [6,7], Axtell, Baeth, and Stickles present several nice results about factorization properties of domains based on their associated irreducible divisor graphs.…”

Section: Introductionmentioning

confidence: 98%

Generalized Irreducible Divisor Graphs

Mooney

2014

Communications in Algebra

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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.

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“…p is a prime in Z by [11,Lemma 2.3]. However, (1, p) is never an essential divisor in any U-factorization of (1, 0) (or τ d -U-factorization or τ d -U-factorization).…”

Section: Irreducible Divisor Graphsmentioning

confidence: 99%

Τ -U-Factorizations AND THEIR GRAPHS IN COMMUTATIVE RINGS WITH ZERO-DIVISORS

Axtell¹,

Mooney²

2015

International Electronic Journal of Algebra

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Abstract. Let R be a commutative ring with identity, and let τ be a relation on the nonzero, non-unit elements of R. In this paper we generalize the definitions of a factorization and a U -factorization via a relation τ and construct a variety of graphs based on these generalizations. These graphs are then examined in an effort to determine ring-theoretic properties.Mathematics Subject Classification (2010): 13A99, 05C25

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Irreducible Divisor Graphs in Commutative Rings with Zero Divisors

Cited by 12 publications

References 9 publications

Irreducible Divisor Graphs and Factorization Properties of Domains

Irreducible Divisor Graphs and Factorization Properties of Domains

Generalized Irreducible Divisor Graphs

Τ -U-Factorizations AND THEIR GRAPHS IN COMMUTATIVE RINGS WITH ZERO-DIVISORS

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