Generalized U-factorization in commutative rings with zero-divisors

Mooney, Christopher

doi:10.1216/rmj-2015-45-2-637

Cited by 8 publications

(32 citation statements)

References 23 publications

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“…As in [16,Corollary 3.3], with τ a symmetric relation, any τ -factorization can be rearranged to form a τ -U-factorization. By (1), this is a τ -U-factorization as desired.…”

Section: Factorizationsmentioning

confidence: 99%

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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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“…As in [16,Corollary 3.3], with τ a symmetric relation, any τ -factorization can be rearranged to form a τ -U-factorization. By (1), this is a τ -U-factorization as desired.…”

Section: Factorizationsmentioning

confidence: 99%

“…It is well known that the associate relations coincide in a présimplifiable ring or when dealing with regular elements, see [17,18].…”

Section: Ring a Ring In Which Every Principal Ideal Is Projectivementioning

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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“…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].…”

Section: Introductionmentioning

confidence: 99%

“…By using τ -factorization, the authors were able to consolidate much of the research in factorization in integral domains into a single study. 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.…”

Section: Introductionmentioning

confidence: 99%

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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“…This research was continued by various authors (e.g., [1,2,3,4,5,6,10,11,15,32]), but in comparison to the domain case our knowledge on the arithmetic of rings with zero divisors is still very rudimentary. One possible approach is to focus on the monoid of regular elements, which definitely makes sense if the ring has few zero divisors.…”

Section: Introductionmentioning

confidence: 99%

ON v-MAROT MORI RINGS AND C-RINGS

Geroldinger¹,

Ramacher²,

Reinhart³

2015

Journal of the Korean Mathematical Society

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Abstract. C-domains are defined via class semigroups, and every Cdomain is a Mori domain with nonzero conductor whose complete integral closure is a Krull domain with finite class group. In order to extend the concept of C-domains to rings with zero divisors, we study v-Marot rings as generalizations of ordinary Marot rings and investigate their theory of regular divisorial ideals. Based on this we establish a generalization of a result well-known for integral domains. Let R be a v-Marot Mori ring, R its complete integral closure, and suppose that the conductor f = (R : R) is regular. If the residue class ring R/f and the class group C( R) are both finite, then R is a C-ring. Moreover, we study both v-Marot rings and C-rings under various ring extensions.

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Generalized U-factorization in commutative rings with zero-divisors

Cited by 8 publications

References 23 publications

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

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

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

ON v-MAROT MORI RINGS AND C-RINGS

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