Zero-Divisor Graphs of Polynomials and Power Series Over Commutative Rings

Axtell, Michael; Coykendall, James; Stickles, Joe

doi:10.1081/agb-200063357

Cited by 88 publications

(49 citation statements)

References 8 publications

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“…In [5] D.F.Anderson, A.Badawi studied connectedness of Total graph of the idealization R(+)M and also investigate diameter and has proved some results on girth of Total graphs. Different aspects of the idealization are thoroughly investigated in [10], [11]. In this paper also extend the study of D.F.Anderson, and A.Badawi .…”

Section: Introductionmentioning

confidence: 68%

Total Graphs of Idealization

EswaraRao¹,

Bharathi²

2014

IJCA

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Section: Introductionmentioning

confidence: 68%

Total Graphs of Idealization

EswaraRao¹,

Bharathi²

2014

IJCA

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“…we can conclude that the diameter and girth of any hypergraph H 3 (R) containing a cycle and satisfying the conditions in the above theorem are bounded by 4 and 9, respectively. Note that a similar result for a zero-divisor graph Γ(R) is studied in [5,8,9,14] as follows. Proof.…”

Section: -Zero-divisor Hypergraphsmentioning

confidence: 94%

Thek-Zero-Divisor Hypergraph of a Commutative Ring

Eslahchi

Rahimi

2007

International Journal of Mathematics and Mathematical Sciences

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The concept of the zero-divisor graph of a commutative ring has been studied by many authors, and the k-zero-divisor hypergraph of a commutative ring is a nice abstraction of this concept. Though some of the proofs in this paper are long and detailed, any reader familiar with zero-divisors will be able to read through the exposition and find many of the results quite interesting. Let R be a commutative ring and k an integer strictly larger than 2. A k-uniform hypergraph H k (R) with the vertex set Z(R,k), the set of all k-zerodivisors in R, is associated to R, where each k-subset of Z(R,k) that satisfies the k-zerodivisor condition is an edge in H k (R). It is shown that if R has two prime ideals P 1 and P 2 with zero their only common point, then H k (R) is a bipartite (2-colorable) hypergraph with partition sets P 1 − Z and P 2 − Z , where Z is the set of all zero divisors of R which are not k-zero-divisors in R . If R has a nonzero nilpotent element, then a lower bound for the clique number of H 3 (R) is found. Also, we have shown that H 3 (R) is connected with diameter at most 4 whenever x 2 / = 0 for all 3-zero-divisors x of R. Finally, it is shown that for any finite nonlocal ring R, the hypergraph H 3 (R) is complete if and only if R is isomorphic to Z 2 × Z 2 × Z 2 .

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“…Zero-divisor graphs have received a lot of attention (see [1,2,4,6,13]), because they are helpful for revealing the ring-theoretic properties via their graph-theoretic properties. In 1988, I. Beck first introduced the concept of zerodivisor graphs of commutative rings in [7], where all elements of a commutative ring R are defined to be vertices and distinct vertices x and y are adjacent if and only if xy = 0.…”

Section: Introductionmentioning

confidence: 99%