A Zero Divisor Graph Determined by Equivalence Classes of Zero Divisors

Spiroff, Sandra; Wickham, Cameron

doi:10.1080/00927872.2010.488675

Cited by 82 publications

(55 citation statements)

References 10 publications

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“…Zero divisor graph is very useful to find the algebraic structures and properties of rings. Recently, the graph of equivalence classes of zero divisors of a commutative Noetherian ring was studied by Spiroff and Wickham in [14]. Note that if R is a unit-duo ring, then for all x, y ∈ R, [xy] = [x][y], i.e., multiplication is well defined on {[x] | x ∈ R}.…”

Section: R[x]mentioning

confidence: 99%

Unit-Duo Rings and Related Graphs of Zero Divisors

Cheol¹,

Lee²,

Park³

2016

Bulletin of the Korean Mathematical Society

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Abstract. Let R be a ring with identity, X be the set of all nonzero, nonunits of R and G be the group of all units of R. A ring R is called[x]r = {xu | u ∈ G}) which are equivalence classes on X. It is shown that for a semisimple unit-duo ring R (for example, a strongly regular ring), there exist a finite number of equivalence classes on X if and only if R is artinian. By considering the zero divisor graph (denoted Γ(R)) determined by equivalence classes of zero divisors of a unit-duo ring R, it is shown that for a unit-duo ring R such that Γ(R) is a finite graph, R is local if and only if diam( Γ(R)) = 2.

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Section: R[x]mentioning

confidence: 99%

Unit-Duo Rings and Related Graphs of Zero Divisors

Cheol¹,

Lee²,

Park³

2016

Bulletin of the Korean Mathematical Society

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“…In [13] the zero divisor graph determined by equivalence classes of zero divisors of a commutative Noetherian ring was introduced as follows.…”

Section: Between ( ) and The Zero Divisorsmentioning

confidence: 99%

Some Properties of the Intersection Graph for Finite Commutative Principal Ideal Rings

Osba

Al-Addasi

AbuGhneim

2014

International Journal of Combinatorics

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Let R be a commutative finite principal ideal ring with unity, and let G(R) be the simple graph consisting of nontrivial proper ideals of R as vertices such that two vertices I and J are adjacent if they have nonzero intersection. In this paper we continue the work done by Abu Osba. We calculate the radius, eccentricity, domination number, independence number, geodetic number, and the hull number for this graph. We also determine when G(R) is chordal. Finally, we study some properties of the complement graph of G(R).

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“…In [12], Redmond introduced the zero-divisor graph with respect to a proper ideal. Since then, there has been a lot of interest in this subject and various papers were published establishing different properties of these graphs as well as relations between graphs of various extensions (see [2], [11], [12] and [13]). Recently, such graphs are used to study semirings [5], [6] and [9].…”

Section: Introductionmentioning

confidence: 99%

A co-ideal based identity-summand graph of a commutative semiring

Atani¹,

Hesari²,

Khoramdel³

2015

Commentationes Mathematicae Universitatis Carolinae

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Abstract. Let I be a strong co-ideal of a commutative semiring R with identity. Let Γ I (R) be a graph with the set of vertices S I (R) = {x ∈ R \ I : x + y ∈ I for some y ∈ R \ I}, where two distinct vertices x and y are adjacent if and only if x + y ∈ I. We look at the diameter and girth of this graph. Also we discuss when Γ I (R) is bipartite. Moreover, studies are done on the planarity, clique, and chromatic number of this graph. Examples illustrating the results are presented.

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A Zero Divisor Graph Determined by Equivalence Classes of Zero Divisors

Abstract: Abstract. We study the zero divisor graph determined by equivalence classes of zero divisors of a commutative Noetherian ring R. We demonstrate how to recover information about R from this structure. In particular, we determine how to identify associated primes from the graph.

Cited by 82 publications

References 10 publications

Unit-Duo Rings and Related Graphs of Zero Divisors

Unit-Duo Rings and Related Graphs of Zero Divisors

Some Properties of the Intersection Graph for Finite Commutative Principal Ideal Rings

A co-ideal based identity-summand graph of a commutative semiring

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