Generalized factorization in commutative rings with zero-divisors

Mooney, Christopher

doi:10.17077/etd.skkt7f53

Cited by 7 publications

(43 citation statements)

References 11 publications

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“…We will say that a is τ -very strongly irreducible or τ -very strongly atomic if a ∼ = a and a has no non-trivial τ -factorizations. See [13] for more equivalent definitions of these various forms of τ -irreducibility.…”

Section: Preliminary Definitions and Resultsmentioning

confidence: 99%

Generalized U-factorization in commutative rings with zero-divisors

Mooney¹

2015

Rocky Mountain J. Math.

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Recently substantial progress has been made on generalized factorization techniques in integral domains, in particular τ -factorization. There has also been advances made in investigating factorization in commutative rings with zero-divisors. One approach which has been found to be very successful is that of U-factorization introduced by C.R. Fletcher. We seek to synthesize work done in these two areas by generalizing τ -factorization to rings with zero-divisors by using the notion of U-factorization.

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Section: Preliminary Definitions and Resultsmentioning

confidence: 99%

Generalized U-factorization in commutative rings with zero-divisors

Mooney¹

2015

Rocky Mountain J. Math.

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“…In [17], the author used the methods established by D.D. Anderson and S. Valdes-Leon in [4] to extend many of the τ -factorization definitions to work also in rings with zero-divisors.…”

Section: Introductionmentioning

confidence: 99%

“…Fletcher in [13,14] and then studied extensively by M. Axtell, N. Baeth, and J. Stickles in [7,8]. In [19], the author studied yet another approach to extending τ -factorization, by using complete factorizations which was touched on in [3] in the case of integral domains.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 3.2, we prove several theorems which describe the relationships between the various τ -regular finite factorization properties that rings may possess. In Section 4, we compare this new method of extending τ -factorization with the previous work in [17] and the relation τ r := τ ∩Reg(R)×Reg(R). In Section 5, we demonstrate how these τ -regular finite factorization properties are related to other finite factorization properties defined in other works, especially [17] and [18].…”

Section: Introductionmentioning

confidence: 99%

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$\tau $-Regular factorization in commutative rings with zero-divisors

Mooney¹

2016

Rocky Mountain J. Math.

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Abstract.Recently there has been a flurry of research on generalized factorization techniques in both integral domains and rings with zero-divisors, namely τ -factorization. There are several ways that authors have studied factorization in rings with zero-divisors. This paper focuses on the method of regular factorizations introduced by D.D. Anderson and S. Valdes-Leon. We investigate how one can extend the notion of τ -factorization to commutative rings with zerodivisors by using the regular factorization approach. The study of regular factorization is particularly effective because the distinct notions of associate and irreducible coincide for regular elements. We also note that the popular U-factorization developed by C.R. Fletcher also coincides since every regular divisor is essential. This will greatly simplify many of the cumbersome finite factorization definitions that exist in the literature when studying factorization in rings with zero-divisors. 2010

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“…See [1] for more information on the developments in the theory of factorization in integral domains. More recently, these concepts have been further generalized by way offactorization in several papers, especially by Anderson and Frazier in [2] as well as the author in [12,13]. In this paper, we seek to take the notion of -factorization and apply it to irreducible divisor graphs.…”

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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Generalized factorization in commutative rings with zero-divisors

Cited by 7 publications

References 11 publications

Generalized U-factorization in commutative rings with zero-divisors

Generalized U-factorization in commutative rings with zero-divisors

$\tau $-Regular factorization in commutative rings with zero-divisors

Generalized Irreducible Divisor Graphs

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