On the complement of a graph associated with the set of all nonzero annihilating ideals of a commutative ring

Visweswaran, S.; Sarman, Patat

doi:10.1142/s1793830916500439

Cited by 8 publications

(12 citation statements)

References 27 publications

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“…Hence, ra ∈ I ⊆ A. This proves that A is an ideal of R. Similarly, it can be shown that B is an ideal of R. Now, it can be shown as in the proof of (i) ⇒ (ii) of ( [19], Proposition 2.10) that both A and B are maximal N-primes of (0) in R and A ∩ B = (0). It is now clear that R is a reduced ring and {A, B} is the set of all minimal prime ideals of R. Proof.…”

Section: Note That a Is Of The Formmentioning

confidence: 57%

“…We know from Lemma 8 that H = (Ω(R)) c . Hence, (Ω(R)) c is star and so, we obtain from (i) ⇒ (ii) of ( [19], Proposition 2.12) that R ∼ = D × F as rings, where F is a field and D is an integral domain but not a field. (ii) ⇒ (i) Let us denote D × F by T , where F is a field and D is an integral domain but not a field.…”

Section: Note That a Is Of The Formmentioning

confidence: 86%

“…Proof. (i) ⇒ (ii) We adapt an argument found in the proof of (i) ⇒ (ii) of ( [19], Proposition 2.10). Let H be a complete bipartite graph with vertex partition V 1 and V 2 .…”

Section: Note That a Is Of The Formmentioning

confidence: 99%

“…As the ideals of a ring also play an important role in studying its structure, M. Behboodi and Z. Rakeei in [10] introduced and investigated an undirected graph called the annihilating-ideal graph of R, denoted by AG(R), whose vertex set is A(R) * and distinct vertices I, J are joined by an edge in AG(R) if and only if IJ = (0). In [10,11], M. Behboodi and Z. Rakeei explored the influence of certain graph-theoretic parameters of AG(R) on the ring structure of R. The annihilatingideal graph of a commutative ring and other related graphs have been studied by several researchers (see for example, [1,2,14,18,19]). Motivated by the work done on the annihilating-ideal graph of a commutative ring, in [2], Alilou, Amjadi and Sheikholeslami introduced and studied a new graph associated to a commutative ring R, denoted by Ω * R , which is an undirected graph whose vertex set is the set of all nontrivial ideals of R and distinct vertices I, J are joined by an edge in this graph if and only if either (Ann R I)J = (0) or (Ann R J)I = (0) (that is, if and only if either Ann R I ⊆ Ann R J or Ann R J ⊆ Ann R I), where for an ideal I of R, the annihilator of I in R, denoted by Ann R I is defined as Ann R I = {r ∈ R : Ir = (0)}.…”

Section: Introductionmentioning

confidence: 99%

“…In [18], we associated and investigated some properties of an undirected graph denoted by Ω(R) whose vertex set is A(R) * and distinct vertices I, J are joined by an edge in Ω(R) if and only if I + J ∈ A(R). In [19], we studied the interplay between the graph-theoretic properties of (Ω(R)) c and the ring-theoretic properties of R. It is useful to recall here that distinct nonzero annihilating ideals I, J are adjacent in (Ω(R)) c if and only if I + J / ∈ A(R). Let H be the subgraph of (Ω * R ) c induced on A(R) * .…”

Section: Introductionmentioning

confidence: 99%

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A new graph associated to a commutative ring

Alilou

Amjadi

Sheikholeslami

2016

Discrete Math. Algorithm. Appl.

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Let [Formula: see text] be a commutative ring with identity. In this paper, we consider a simple graph associated with [Formula: see text] denoted by [Formula: see text], whose vertex set is the set of all nonzero proper ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent whenever [Formula: see text] or [Formula: see text]. In this paper, we initiate the study of the graph [Formula: see text] and we investigate its properties. In particular, we show that [Formula: see text] is a connected graph with [Formula: see text] unless [Formula: see text] is isomorphic to a direct product of two fields. Moreover, we characterize all commutative rings [Formula: see text] with at least two maximal ideals for which [Formula: see text] are planar.

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Section: Note That a Is Of The Formmentioning

confidence: 57%

Section: Note That a Is Of The Formmentioning

confidence: 86%

“…Proof. (i) ⇒ (ii) We adapt an argument found in the proof of (i) ⇒ (ii) of ( [19], Proposition 2.10). Let H be a complete bipartite graph with vertex partition V 1 and V 2 .…”

Section: Note That a Is Of The Formmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A new graph associated to a commutative ring

Alilou

Amjadi

Sheikholeslami

2016

Discrete Math. Algorithm. Appl.

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On the planarity of a graph associated to a commutative ring and on the planarity of its complement

Visweswaran

Sarman

2017

São Paulo J. Math. Sci.

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Some results on the total zero-divisor graph of a commutative ring

Visweswaran

2024

AJMS

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PurposeThe purpose of this paper is to characterize a commutative ring R with identity which is not an integral domain such that ZT(R), the total zero-divisor graph of R is connected and to determine the diameter and radius of ZT(R) whenever ZT(R) is connected. Also, the purpose is to generalize some of the known results proved by Duric et al. on the total zero-divisor graph of R.Design/methodology/approachWe use the methods from commutative ring theory on primary decomposition and strong primary decomposition of ideals in commutative rings. The structure of ideals, the primary ideals, the prime ideals, the set of zero-divisors of the finite direct product of commutative rings is used in this paper. The notion of maximal Nagata prime of the zero-ideal of a commutative ring is also used in our discussion.FindingsFor a commutative ring R with identity, ZT(R) is the intersection of the zero-divisor graph of R and the total graph of R induced by the set of all non-zero zero-divisors of R. The zero-divisor graph of R and the total graph of R induced by the set of all non-zero zero-divisors of R are well studied. Hence, we determine necessary and sufficient condition so that ZT(R) agrees with the zero-divisor graph of R (respectively, agrees with the total graph induced by the set of non-zero zero-divisors of R). If Z(R) is an ideal of R, then it is noted that ZT(R) agrees with the zero-divisor graph of R. Hence, we focus on rings R such that Z(R) is not an ideal of R. We are able to characterize R such that ZT(R) is connected under the assumptions that the zero ideal of R admits a strong primary decomposition and Z(R) is not an ideal of R. With the above assumptions, we are able to determine the domination number of ZT(R).Research limitations/implicationsDuric et al. characterized Artinian rings R such that ZT(R) is connected. In this paper, we extend their result to rings R such that the zero ideal of R admits a strong primary decomposition and Z(R) is not an ideal of R. As an Artinian ring is isomorphic to the direct product of a finite number of Artinian local rings, we try to characterize R such that ZT(R) is connected under the assumption that R is ta finite direct product of rings R1, R2, … Rn with Z(Ri) is an ideal of Ri for each i between 1 to n. Their result on domination number of ZT(R) is also generalized in this paper. We provide several examples to illustrate our results proved.Practical implicationsThe implication of this paper is that the existing result of Duric et al. is applicable to large class of commutative rings thereby yielding more examples. Moreover, the results proved in this paper make us to understand the structure of commutative rings better. It also helps us to learn the interplay between the ring-theoretic properties and the graph-theoretic properties of the graph associated with it.Originality/valueThe results proved in this paper are original and they provide more insight into the structure of total zero-divisor graph of a commutative ring. This paper provides several examples. Not much work done in the area of total zero-divisor graph of a commutative ring. This paper is a contribution to the area of graphs and rings and may inspire other researchers to study the total zero-divisor graph in further detail.

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On the complement of a graph associated with the set of all nonzero annihilating ideals of a commutative ring

Cited by 8 publications

References 27 publications

A new graph associated to a commutative ring

A new graph associated to a commutative ring

On the planarity of a graph associated to a commutative ring and on the planarity of its complement

Some results on the total zero-divisor graph of a commutative ring

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