Irreducible Divisor Graphs

Coykendall, Jim; Maney, Jack

doi:10.1080/00927870601115807

Cited by 24 publications

(43 citation statements)

References 12 publications

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“…The following was a particularly nice result from the integral domain case in [22] which is a generalization of a theorem of [16]. (1) D is a τ -UFD.…”

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

confidence: 99%

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

Mooney¹

2016

International Electronic Journal of Algebra

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Abstract. In this paper, we combine research done recently in two areas of factorization theory. The first is the extension of τ -factorization to commutative rings with zero-divisors. The second is the extension of irreducible divisor graphs of elements from integral domains to commutative rings with zero-divisors. We introduce the τ -irreducible divisor graph for various choices of associate and irreducible. By using τ -irreducible divisor graphs, we find that we are able to obtain, as subcases, many of the graphs associated with commutative rings which followed from the landmark 1988 paper by I. Beck.We then are able to use these graphs to give alternative characterizations of τ -finite factorization properties previously defined in the literature.Mathematics Subject Classification (2010): 13A05, 13E99, 13F15, 05C25

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“…The following was a particularly nice result from the integral domain case in [22] which is a generalization of a theorem of [16]. (1) D is a τ -UFD.…”

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

confidence: 99%

“…We pause to give the definition of the irreducible divisor graph in the standard factorization setting and in atomic integral domains as used in [9,16] …”

Section: General Graph Theoretic and Irreducible Divisor Graph Definimentioning

confidence: 99%

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

Mooney¹

2016

International Electronic Journal of Algebra

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“…The concept of the irreducible divisor graph of an element in a commutative integral domain was introduced in [Coykendall and Maney 2007] and further studied in [Axtell et al 2011]. We now give a similar definition for the irreducible divisor graph of an element in a multiplicative semigroup.…”

Section: Irreducible Divisor Graphsmentioning

confidence: 99%

“…The fundamental result in the theory of irreducible divisor graphs, proved in [Coykendall and Maney 2007;Axtell et al 2011] tells us that an atomic integral domain R is a UFD if and only if G(x) is connected (equivalently, complete) for all nonunits x ∈ R. In fact, the proof of this result goes through for any commutative, cancellative semigroup. As should be no surprise, the examples above give disconnected graphs.…”

Section: Irreducible Divisor Graphsmentioning

confidence: 99%

Untitled

2012

Involve

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The steady states of a mathematical model for the dynamics of Chagas disease, developed by Spagnuolo et al., are studied and numerically simulated. The model consists of a system of four nonlinear ordinary differential equations for the total number of domestic carrier insects, and the infected insects, infected humans, and infected domestic animals. The equation for the vector dynamics has a growth rate of the blowfly type with a delay. In the parameter range of interest, the model has two unstable disease-free equilibria and a globally asymptotically stable (GAS) endemic equilibrium. Numerical simulations, based on the fourth-order Adams-Bashforth predictor corrector scheme for ODEs, depict the various cases.

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“…In this paper, we focus on the notion of an irreducible divisor graph first formulated by Coykendall and Maney in [9]. 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.…”

Section: Introductionmentioning

confidence: 99%

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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Irreducible Divisor Graphs

Cited by 24 publications

References 12 publications

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

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

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Generalized Irreducible Divisor Graphs

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